Pressure on Theresa May to set a firm resignation date has increased ahead of a meeting with the chairman of the powerful 1922 Committee of Tory backbenchers.
Tuesday’s meeting with Sir Graham Brady comes after the 1922 Committee requested “clarity” about the Prime Minister’s timetable for standing down and triggering a leadership contest.
Meanwhile, senior Tory activists will consider the question of Mrs May’s leadership at an emergency meeting of association chairmen next month.
A revolt against Mrs May could become more likely if talks with Labour result in a Brexit compromise which would be unacceptable to Tory Eurosceptics.
Government ministers and their Labour counterparts will resume talks in Westminster aimed at breaking the deadlock.
Foreign Secretary Jeremy Hunt said both sides need to be willing to compromise.
Mr Hunt said that after both main parties lost ground in the local elections in England, it was a “crucial week” coming up for the Brexit negotiations.
However, he said that he did not believe a permanent customs union with the EU - supported by Labour - offered a “sustainable, long-term solution” to the current impasse.
“I personally think that any kind of permanent customs union wouldn’t work in the long run because our economy is too big, but let’s see what the parties come up with,” he told BBC Radio 4’s Today programme.
Anger within the Tory ranks at Mrs May has been fuelled by the potential for a softer Brexit deal with Labour.
Leading Eurosceptic Sir Bill Cash told the Press Association: “The time has come for her to resign.
“She needs to be given a date. The sooner the better. But it needs to be done in an orderly manner.”
Grassroots Tories will hold a no-confidence vote in Mrs May on June 15.
The vote by members at an EGM of the National Conservative Convention would not be binding, but would add pressure on Mrs May to quit if passed.
In a message to members of the convention, reported by the Conservative Home website, chairman Andrew Sharpe said they would be asked to vote on a motion stating that “we no longer feel that Mrs May is the right person to continue as Prime Minister to lead us forward in the negotiations” and “therefore with great reluctance ask that she considers her position and resigns”.
Mrs May has said she will step down if her Withdrawal Agreement is ratified, but - with the deadline for Brexit extended to the end of October - has not made clear how long she intends to stay if no deal is reached.
Sir Geoffrey Clifton-Brown, treasurer of the 1922 Committee, put the spotlight on a departure date by insisting Mrs May announce a “road map” for her resignation after the European elections set for May 23.
Brexit Party leader Nigel Farage told the Press Association: “I’m amazed she is still there.”
The resumption of Government talks with Labour comes amid increased tension between the two sides.
Shadow chancellor John McDonnell said he had no trust in Mrs May after reports emerged that she is ready to offer a temporary customs arrangement with the EU to the opposition.
Labour accused Mrs May of having “blown the confidentiality” of the talks.
Party leader Jeremy Corbyn said Brexit had a big impact on last week’s local election results, which saw huge losses for the Tories and a disappointing showing for Labour.
A number of senior Tory Brexiteers have said they would not vote for a customs union.
Referring to talks with Labour, children’s minister Nadhim Zahawi said: “Currently, if you look at the Withdrawal Agreement, the customs arrangements, or the alignment with the EU, would go on until December 2021, if Labour believe, actually, that they would rather go to the next general election, which is 2022, for example, then actually, that’s still a temporary customs arrangement.
“And then, whoever is leader of the Conservative Party can then lay out their stall as to the next instalment of negotiations.”