First Minister Nicola Sturgeon and Prime Minister Theresa May braced for second Scottish independence referendum showdown

Meanwhile, Downing Street suggested the two leaders could meet before Mrs May triggers the Article 50 withdrawal process from the EU at the end of this month.

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The Prime Minister’s spokesman said: “There will be opportunities between now and the triggering of Article 50 for more conversations to take place.”

Mrs May is preparing a grand tour of the UK before sending her Article 50 letter to Brussels, which would be likely to include a visit to Edinburgh.

The Herald understands one possibility is for Mrs May and Ms Sturgeon to meet during the Scotland leg.

A failure to confer would worsen already strained cross-border relations.

A Scottish Government source said there had not yet been a direct approach from Whitehall but the First Minister was keen to sit down with Mrs May: “Just name the time and place.”

The hint of a rapprochement follows one of the most torrid weeks for devolution since 1999.

Ms Sturgeon announced on Monday that she intended to hold a referendum to let Scots choose between Brexit in the UK and independence and a different relationship with Europe.

She insisted she had a “cast iron mandate” to call the vote, as the SNP manifesto said Holyrood should have the right to hold one if Scotland was taken out the EU unwillingly.

Mrs May’s response was to rule out a referendum on Ms Sturgeon’s timetable by denying Holyrood the necessary transfer of powers, arguing the UK government’s focus needed to be on the two years of complex Brexit negotiations that lie ahead instead.

The Scottish Tory leader Ruth Davidson later said Ms Sturgeon would have to “earn the right” to another referendum, implying she would need to win another Holyrood majority.

Signalling her readiness to bend, Ms Sturgeon said: “I have set out when I think it would be right. She [Mrs May] does not agree with that.

“So let her set out when she thinks it would be right, and then let’s have a discussion about it.

“Who knows, we might only be a matter of weeks or months apart.

“So I’m up for continued discussion, but .. you can’t have discussion and reach compromise with people who are not prepared to enter into discussion and are not prepared to countenance compromise, and that so far has been my experience of the Prime Minister.”

Ms Sturgeon will repeat her offer in her keynote speech to the SNP conference today.

She will tell activists in Aberdeen that if the Prime Minister refuses to accept next week’s Holyrood vote on a transfer of referendum powers, it would “shatter beyond repair any notion of the UK as a respectful partnership of equals”.

She will say of Mrs May: “She has time to think again and I hope she does. If her concern is timing then - within reason - I am happy to have that discussion.”

Scottish Tory chief whip John Lamont said: “Nicola Sturgeon said as late as Thursday that a referendum after April 2019 would be 'too late'. Now she has changed her mind and appears to be trying to engage in some kind of horse-trading with the UK Government. The future of Scotland is not a game. It is time for the First Minister to act responsibly over our future."

The first day of SNP conference also saw a row break out over whether Ms Sturgeon might call a unilateral referendum without Westminster consent if Mrs May continued to block her.

In an interview with STV, the First Minister refused nine times to rule out a consultative referendum, while her deputy John Swinney failed three times to rule one out on Radio 4.

Asked repeatedly if a consultative referendum was on the table, Ms Sturgeon said there were "various options”, but insisted: "My intention is that we have a referendum on the same basis we had as the one in 2014."

The Catalans held a consultative referendum on self-determination in 2014, in defiance of the Spanish government, leading to president Artur Mas being banned from public office for life.

Alex Salmond flirted with a consultative referendum in the 2007-11 parliament, but ministers dropped the idea after learning it would be challenged in court and end in a legal quagmire.

Labour MP Ian Murray said the SNP “must immediately withdraw the threat to impose an illegitimate and divisive referendum”.

He said: "The SNP described the Edinburgh Agreement as the gold standard, because it established a legal basis for the 2014 referendum. It would be entirely unacceptable for a similar approach not to be taken if there is another referendum. Credible governments and political parties do not impose referendums that do not stand up to legal scrutiny.”

Deputy Scottish Tory leader Jackson Carlaw said: “Such a process would make Scotland the laughing stock of the world. The 2014 referendum was legal and fair, but a wildcat version – as so many SNP members seem to want – would be anything but. It appears Nicola Sturgeon's half-baked SNP's referendum plans are descending into chaos.”

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Scottish LibDem leader Willie said: “Nicola Sturgeon’s failure to rule out a wildcat referendum is reckless. It would be following the lead taken by the Catalans who organised a referendum that was boycotted by the opposition and led to nothing. An illegal referendum is plain wrong.”

Scottish Government advisors insisted there was no intention to hold a rogue vote, and ministers wanted to follow the precedent of 2014 and have a “legal, fair and decisive” ballot based on Westminster passing a so-called Section 30 order.

At the Conservative Spring Forum in Cardiff, Mrs May accused the SNP of proposing "muddle on muddle" and using Brexit as pretext to engineer a second referendum.

Mrs May insisted the Tory Party believed "heart and soul" in “this precious, precious Union” and claimed the Conservatives represented the "new centre ground" of British politics.