The tape revealed that Mrs Miller's special adviser Jo Hindley "flagged up" the fact that the Cabinet minister would be meeting the paper's then editor about the Leveson report into press standards - something which then editor Tony Gallagher interpreted as a threat.
Meanwhile, details of letters sent by Mrs Miller to the parliamentary commissioner investigating her expenses claims led to allegations that the MP may have bullied the watchdog during the lengthy inquiry.
Commissioner Kathryn Hudson recommended that Mrs Miller should repay £45,000 in expenses for a house which she shared with her parents, but the cross-party House of Commons Standards Committee overruled the watchdog and decided she only needed to hand back £5,800 in overclaimed mortgage interest.
Mrs Miller, who made a 32-second apology to the Commons earlier this week, has won the "warm support" of Prime Minister David Cameron.
But Labour MPs have urged police to examine Mrs Miller's expenses and demanded the publication of the full minutes of the committee's meetings to discuss the case behind closed doors.
The 10-member committee attempted to defuse criticism of its actions by releasing a joint statement with Ms Hudson describing the situation as "complex" and stressing Mrs Miller had been rebuked for failing to co-operate with the investigation.
In the audio tape released by the Telegraph, Ms Hindley tells the reporter that another journalist who called at Mrs Miller's home had spoken to the MP's father, who had recently been in hospital. Some details were redacted from the recording to protect his privacy.
"I should just flag up as well, whilst you're on it, that when she doorstepped him, she got Maria's father, who's just had a [redacted] and come out of [redacted]," said Ms Hindley.
"Maria has obviously been having quite a lot of editors' meetings around Leveson at the moment. So I'm just going to flag up that connection for you to think about."
Explaining why he regarded the reference to Leveson as a veiled threat, Mr Gallagher, told BBC2's Newsnight: "Bear in mind this was a time of what I would call anti-press hysteria.
"The press was feeling very vulnerable just after the publication of the Leveson Report and there was a great desire on the part of all media organisations not to fall foul of somebody raising the spectre of Leveson.
"We were in no doubt that threats were being made. Joanna Hindley was not even attempting to be sophisitcated about it - she menaced the reporter openly... The reporter took that as a very clear threat."
But a Whitehall source said that it was clear from the recording, made without Ms Hindley's knowledge, that she had been voicing concerns about the doorstepping of Mrs Miller's father, who was ill, and had been making clear that the minister would raise the issue at an upcoming meeting with the editor, which happened to relate to the Leveson report.
The source said that Mrs Miller did in fact speak to Mr Gallagher about the treatment of her father, and received a written apology from him.
Mr Gallagher claimed that he later received a similar call from Downing Street director of communications Craig Oliver, adding: "When you get the Prime Minister's spokesman making a similar phone call to you a couple of days later, you add all three calls up and you can only conclude that they are trying to harass you and stop you from publishing the story."
But Mr Oliver told the BBC: "It is utterly false to suggest that I threatened Tony Gallagher over Leveson in any way. The conversation I had with him was about the inappropriate doorstepping of an old man."
Emails released by the Standards Committee showed how Mrs Miller tried to persuade Ms Hudson that her investigation into her expenses was "irrational, perverse and unreasonable" and suggested that she might go over the watchdog's head to ask the cross-party group of MPs to intervene.
In one email, Mrs Miller said: "It may be that I shall need to refer this to the supervisory jurisdiction of the standards committee but I hope this can be avoided."
In another, she wrote: "As should by now be obvious, a decision to uphold the complaint would be irrational, perverse and ... unreasonable - that is to say would be a decision that no reasonable decision maker could properly reach."
And in another message, she wrote: "In light of the evidence that is before you ... to continue to regard this spurious complaint as a serious matter would give it credence it does not deserve and undermine the inquiry process in comparison to issues that really are serious matters."
Mr Mann, the MP whose complaint sparked the Commissioner's investigation, said: "These emails show that Maria Miller bullied and threatened the independent Commissioner.
"This is a gross abuse of her position as a member of the Cabinet. This issue alone warrants her resignation or immediate dismissal."
Labour MP Tommy Docherty wrote to the Metropolitan Police asking them to look at whether the commissioner's report raised any issues of potential criminal behaviour.
Speaking on a visit to Devon, the Prime Minister said: "It was found she had made mistakes, she accepted that, repaid the money, she apologised unreservedly to the House of Commons so I think that we should leave it there."
The inquiry was launched in December 2012 following allegations that Mrs Miller used some £90,000 in expenses between 2005 and 2009 to fund a house where she lived with her parents.
In a joint statement released today, committee chairman Kevin Barron and Ms Hudson said: "As both the Commissioner's memorandum and the Committee's report make clear, this case dealt with complex matters relating to the rules as they were nearly a decade ago.
"Both the Commissioner's memorandum and the Committee's report set out in full the reasoning behind their conclusions, and the evidence which they used in coming to those conclusions."
Ukip leader Nigel Farage said David Cameron had made a "disastrous error of judgment" over the way he has handled the issue.
During a phone-in broadcast on The Telegraph website, he said: "I think that Cameron should have asked her to resign and if she hadn't resigned, he should have kicked her out."
Labour leader Ed Miliband told the Telegraph that Mr Cameron had shown "weak leadership" and should have "stood up and said that what she did was wrong".
The paper said in a leader column: "Let us not forget that Mrs Miller is also the minister in charge of future press regulation. The accusation has been made that this fact was used to make journalists back away from reporting on her expenses, which would be an indictment of the influence that press regulation by statute could have over free speech.
"If our MPs cannot hold an honest conversation about regulation of their own expenditure, how can we expect them to hold an honest conversation about the maintenance of the free press?"
The Telegraph editorial added: "Questions about Mrs Miller's actions are unlikely to go away and many people will be asking themselves why the Prime Minister is standing so doggedly by her.
"Whatever Mr Cameron's motivation, he must surely understand that voters share his ambition for a cleaner politics and are willing him to take appropriate action. Alas, the expenses scandal is a manifestation of the lingering feeling that the gulf between the electors and the elected remains sizeable."
But Conservative MP Therese Coffey, who sat on the Commons Culture, Media and Sport Committee from 2010-12, that she did not believe the recorded telephone conversation amounted to a threat to the paper.
Dr Coffey told Newsnight: "That's not what I heard. I heard an adviser to the Culture Secretary suggesting that it's inappropriate to be doorstepping elderly parents of somebody who has just come out of hospital.
"The issue about Leveson has been going on for some time, but I have not seen any change in newspapers' attitudes in wanting to publish what they wish, and I think that's the right thing to do - I believe in press freedom. It's right for them to pursue avenues, but do that appropriately."
But freedom of information campaigner Heather Brooke, whose work led to the original exposure of the expenses scandal, told the programme: "When I heard that, it does strike you as a threat. If you're a reporter and somebody calls you up, mentions that they are involved in a very serious inquiry about regulating the press and then mentions that they are going to talk to your bosses, then that is something that you would as a reporter take as a method of intimidation to try to shut you out of the investigation you were trying to conduct."
The former chairman of the Committee on Standards in Public Life, Sir Alistair Graham, said Mrs Miller's failure to co-operate fully with the commissioner's inquiry was "pretty shocking".
Sir Alistair told the BBC Radio 4 Today programme: "The degree of lack of co-operation or the attempt to divert the commissioner from addressing the issues concerned seems fairly exceptional. I think particularly for a senior Cabinet minister, who you expect to show a leadership role in co-operating with whatever expenses system is around, it is pretty shocking.
"I think the public will be very shocked that the committee did overturn one of the key recommendations about how much should be repaid back, when there is a real possibility that the minister made a capital gain with the help of public funds."
Sir Alistair and the CSPL's current chair, Lord Bew, each raised questions about the position of the non-politicians - known as "lay members" - appointed to the Commons Standards Committee in the wake of the expenses scandal.
The CSPL's recommendation that they should have full voting rights was not accepted by MPs, and the three lay members do not have a vote, but play a full part in discussions and have the power to append an opinion to any committee report.
In a statement to the Today programme, Lord Bew said: "The committee's expenses inquiry in 2009 looked at the disciplinary systems within the House of Commons and recommended that the membership of the Commons Standards and Privileges Committee should include lay members with full voting rights to better represent the views of the public.
"We are pleased that lay members have been appointed to that committee. However, there remains some debate about whether they will have full voting rights."
Sir Alistair said: "I think what the Maria Miller case highlights is the need for further reform, because if you do a comparison between the disciplinary committee system for Members of Parliament with other professional groups like lawyers, doctors, dentists, opticians, they have an independent lay chair and they have a majority of lay voting members on the committee, supplemented normally by one or two people from the profession itself.
"That seems to be a more appropriate model for the House of Commons. Plus the fact that those hearings are held in public."
Lord Bew declined to comment on Mrs Miller's case, but added: "At a time when we are beginning to move on from the damaging expenses scandal, Parliament really needs to be seen to be listening and not repeating past mistakes."