Lawyers for the Tory peer said they were taking legal action as a result of the programme, which led to the resignation of BBC director-general George Entwistle on Saturday.
In an interview being broadcast today, he said the BBC could have saved "a lot of agonising and money" by simply calling him before the programme went out.
Ofcom said today it was investigating the broadcast, which has fuelled a crisis at the corporation and a revamp of the "chain of command" which had been operating in BBC news since the Jimmy Savile scandal broke.
And the regulator is also investigating ITV1's This Morning after presenter Phillip Schofield brandished a list of names of alleged abusers he had found on the internet and handed it to the Prime Minister during a live interview, asking if he would investigate them.
The stunt provoked fury last week and ITV said today that disciplinary action had been taken.
Lord McAlpine was mistakenly implicated by Newsnight's November 2 broadcast in a paedophile ring that targeted children at a care home in Wrexham in North Wales.
Although the programme did not name the peer - referring to only a senior Conservative from the Thatcher era - he was quickly identified online.
Lord McAlpine said the incident had left him devastated and it got into his "soul".
He told BBC Radio 4's The World At One programme: "They could have saved themselves a lot of agonising, and money actually, if they had just made that telephone call.
"They should have called me and I would have told them exactly what they learned later on - that it was complete rubbish and that I had only ever been to Wrexham once in my life."
He added: "It gets into your bones, it makes you angry, and that's extremely bad for you to be angry, and it gets into your soul, and you just think there is something wrong with the world."
Lord McAlpine's solicitor Andrew Reid told Radio 4: "Lord McAlpine is more than aware that the ultimate people who will be paying for any monies that he may receive are, in fact, the licence-payers, the people who really own the BBC, and he is very much aware of this and hence any agreement that is reached is tempered in the light of that."
Newsnight carried a full, on-air apology for the broadcast a week later. And an official report into the botched investigation by the BBC's Scotland director Ken MacQuarrie concluded Newsnight staff failed to complete "basic journalistic checks".
Mr MacQuarrie also found there was confusion about who had the ultimate responsibility for "final editorial sign-off", adding that the programme's editorial management structure had been "seriously weakened" as a result of the editor having to step aside over the Savile scandal, and the departure of the deputy editor.
Although legal advice was sought over the report, no right of reply was offered to the unnamed individual at the centre of the allegation.
A BBC spokeswoman said: "The BBC is hopeful that it can agree a settlement with Lord McAlpine today."
The programme featured an interview with Steve Messham, an abuse victim who said a senior political figure of the time abused him. He later said he wrongly identified his abuser and apologised.
The botched programme heaped more pressure on the BBC, which has been left reeling from the Savile abuse sex scandal.
A review led by former Sky News chief Nick Pollard is already looking into an earlier decision to shelve a Newsnight investigation into the late television presenter's sexual abuse of youngsters.
BBC director of news Helen Boaden and her deputy Stephen Mitchell have stepped aside from their posts temporarily in the wake of the Savile affair.
The corporation said this was in response to the "lack of clarity" surrounding who is in charge while the Pollard Review is making its inquiries.
The BBC's head of news gathering Fran Unsworth and Ceri Thomas, editor of the Radio 4 Today programme, are filling in for Ms Boaden and Mr Mitchell until the review is concluded.
Acting director general Tim Davie, who took charge of the BBC following the resignation of Mr Entwistle, vowed on Monday to "get a grip of the situation".
The corporation has also launched an investigation into the culture and practices at the BBC during Savile's career and another into the handling of past sexual harassment claims.
Solicitor Mr Reid said those who had named Lord McAlpine on the internet should come forward to settle the matter quickly and avoid further legal issues. He suggested many of them have already been identified.
Speaking in the World At One interview, he said: "What we're basically saying to people is, look, we know - in inverted commas - who you are, we know exactly the extent of what you've done.
"It's easier to come forward and see us and apologise and arrange to settle with us because, in the long run, this is the cheapest and best way to bring this matter to an end."
Sally Bercow, wife of House of Commons speaker John Bercow, is among those who alluded to Lord McAlpine on Twitter. She subsequently apologised.
Lord McAlpine said in Radio 4's World At One interview today that the damage "can't be repaired" and he had been left with the legacy of suspicion.
He said that his legal team will ensure that anyone who brings the matter up again will be "very, very foolish".
And he said the suggestion of being a paedophile - which is with him constantly - is one of the worst things of which anyone could be accused.
"I don't want to be too dramatic about the thing, but Boris (Johnson) got it right. There is nothing as bad as this that you can do to people," Lord McAlpine said.
"Because they are quite rightly figures of public hatred. And suddenly to find yourself a figure of public hatred, unjustifiably, is terrifying."
Asked about his reputation, he said: "No it can't be repaired. It can be repaired to a point, but there is a British proverb which is insidious and awful where people say 'there's no smoke without a fire', you know, 'he appears to be innocent, but..'."
He went on: "It's very difficult and so this is the legacy that sadly the BBC have left me with."
Asked whether he felt he would recover from the incident, he said: "Well look, I'm 70 years old, I've got a very dicky heart. And so I don't want to die. Not for another 20 years at least.
"But I don't see it going away completely. I think in the light of the arrangements that I can make, my lawyer will make, anyone who does bring it up is going to be very, very foolish.
"I will feel better not from settlements, but from my wife, my family, my friends. The people I don't even know who've written to me. That's what makes you feel better. The settlement will just be a warning - don't go there."
Lord McAlpine said the accusation had also been damaging to his family, but they had supported him.
"I was very lucky - I've got a fine wife who was very good about this whole thing and very helpful in dealing with it.
"Of course she was upset, she was furious. But you know I have a big family, most of whom are engaged in business or charitable undertakings or a whole range of things. They're well known. This was as damaging to them as it was to me."
Lord McAlpine, who said he did not want to meet his accuser, said the reality of the situation was "beginning to sink in".
"But even now, when we go away and I go to bed for the night and I go to sleep, I wake up in the morning and I'll still wonder about this thing," he said.
"It'll still be on my mind. It becomes part of your conversation, it becomes part of your life."