The push to topple Theresa May ahead of party conference appears to be gathering strength.
Graham Brady, the chairman of the powerful 1922 Committee of Tory backbenchers, remained silent following a meeting with the Prime Minister about setting a firm resignation date.
Mr Brady will update committee members on Wednesday following angry calls for "clarity" on Mrs May's timetable for standing down and triggering a leadership contest.
The treasurer of the 1922 Committee, Sir Geoffrey Clifton-Brown, told the Press Association he would back a grass roots bid to force Mrs May out with a no-confidence vote next month if she did not go of her own volition.
The Tory MP for the Cotswolds said it was the Prime Minister's decision on when she should go but he would "absolutely" support grass roots moves to try to topple her in June if her departure date was not imminent.
When asked what should happen if Mrs May failed to set out her own timetable for departure, he said: "It begins to get much more messy.
"It would be much easier and I think the European elections would be much easier if she did set out her own timetable to go but it is up to her.
"I think it's quite possible they (grass roots members) might vote for no-confidence in June."
Senior Tory activists will consider the question of Mrs May's leadership at an emergency meeting of association chairmen – set for Saturday May 15.
Tory MPs are powerless to remove Mrs May following a failed bid to boot her out as leader in December last year but the grass roots are calling for a rule change.
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 "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".
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."
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.
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 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.
But Downing Street rejected the claim, with Mrs May's official spokesman saying: "I don't believe that is the case. We have preserved the confidentiality of the talks."
Cabinet discussed the Brexit talks for around an hour on Tuesday morning, the spokesman said.
"The Prime Minister said that while an agreement with the Opposition had not been reached, the public had sent a clear message in the local elections that they want both of the main parties to get on with delivering Brexit," he said.
"She said it was right to continue with the talks process."
The spokesman said at Cabinet "there was a broad understanding that there is a need to get on with this".
Last week's local elections, which had disastrous results for the Conservatives and disappointing losses for Labour, have increased pressure on both sides for a breakthrough in the cross-party talks, which have run on for over a month without producing agreement.
No formal deadline has been set for the conclusion of talks.
But it is thought likely that if no progress is made this week, the Government may move towards a process of indicative votes on Brexit options in the Commons.
Mrs May has made clear she would like both the Government and Labour to commit to respecting the outcome of any such votes.
The Prime Minister's spokesman said: "What is absolutely vital is that we find a way forward which can command a stable majority in the House of Commons.
"That's what the Prime Minister is working to achieve."
Mrs May's effective deputy David Lidington was leading the Government side in Tuesday's Brexit talks at the Cabinet Office, with Brexit Secretary Stephen Barclay, chief whip Julian Smith, Environment Secretary Michael Gove and Number 10 chief of staff Gavin Barwell also expected to be in attendance.