Senior Conservatives are expected to hold further discussions on whether to change the party rules to enable an early leadership challenge to Theresa May.
Officers of the Tory backbench 1922 Committee met in Westminster on Tuesday amid growing pressure for the Prime Minister to name the date of her departure.
Afterwards there was no formal statement from the meeting which was said have been inconclusive.
The meeting venue was switched at the last minute to enable members to avoid waiting reporters.
However it is reported that there will be further discussions ahead of the weekly meeting of the full 1922 Committee on Wednesday.
Health Secretary Matt Hancock warned that changing the prime minister would not resolve the deadlock in Parliament over Brexit.
Speaking at a TaxPayers’ Alliance launch event in London, he said he still hoped it would be possible to get a majority for a deal.
“Changing the prime minister will not change what we need to do to deliver Brexit,” he said.
“I hope the House of Commons will come to a majority to be able to deliver the result on the referendum.”
Ahead of Tuesday’s talks, the committee chairman Sir Graham Brady met privately with Mrs May, when he is reported to have told her MPs want her to announce when she is going.
Under current party rules, MPs cannot mount a fresh leadership challenge until 12 months after last December’s failed attempt.
However, amid growing frustration over the latest delays to Brexit, some MPs now want to the rules to be rewritten to allow another challenge as early as June.
One member of the executive, joint executive secretary Nigel Evans, has called publicly for Mrs May to go “as soon as possible”.
However other members were reported to have pushed back at Tuesday’s meeting, questioning what a fresh leadership contest at the current time would achieve.
Mrs May has already sought to buy time, promising Tory MPs last month that she would go once she has delivered Brexit.
But following the latest extension to the Article 50 withdrawal process there is growing impatience among her critics.
They fear the Tories will suffer heavy losses to Nigel Farage’s Brexit Party if – as now seems likely – the UK is forced to go ahead with voting in the European elections on May 23.
Among the potential candidates for Mr Farage’s new outfit will be former shadow home secretary Ann Widdecombe, who told the Daily Express: “The public needs to send a very clear message and that is we expect the vote to be respected so just get on with the job of getting us out of the EU.”
Meanwhile, cross-party talks between the Government and Labour aimed at forging a common way forward are continuing amid recriminations at the slow pace of progress.
After talks resumed in Whitehall on Tuesday following the Easter break, Chancellor Philip Hammond is due to have a further meeting with his Labour counterpart John McDonnell on Wednesday.
However the Prime Minister’s official spokesman said that while the discussions with Labour were “serious” they were proving “difficult” in some areas, and that progress was needed “urgently” to enable Britain to leave the EU as soon as possible.
Labour leader Jeremy Corbyn, however, put the blame for lack of progress on the Government’s refusal to shift on its “red lines”.
“We’ll continue putting our case but quite honestly there’s got to be change in the Government’s approach,” he said.
“They cannot keep on just regurgitating what has already been emphatically rejected three times by Parliament, there’s got to be a change.”