Theresa May has bought herself another week’s grace as prime minister, hinting she will bring the EU withdrawal bill to parliament before the European elections and promising to meet a powerful backbench committee who have demanded that she set out her timetable for stepping down.

After a fortnight of furious demands by Tory MPs that she give a firm date for her departure, Sir Graham Brady, the chair of the 1922 Committee, said May had agreed to meet him and the 13-strong executive of Tory backbenchers next week.

He gave no indication that May intended to provide a firm departure date at the meeting, but the promise will buy the prime minister an extra week to continue cross-party Brexit talks with Labour, before the Conservatives could consider changing leadership rules to force her exit.

Brady, who has met with May twice over the past few weeks, said he was convinced she would now attempt to pass Brexit legislation as a way of ratifying the deal – though it is unclear how it could command a majority.

Q&A What is the 1922 Committee? Show Hide The 1922 Committee is the shorthand name for the parliamentary group of backbench Conservative MPs that meets weekly while parliament is in session. The group has an executive body of 18 backbench MPs who oversee the organisation of Conservative party leadership elections. The name derives from it being formed originally by a small group of MPs who were elected for the first time in the 1922 general election, before expanding to become the main representative group of Conservative MPs who are not part of the government itself. Photograph: Conservative Party/AFP

“It is the intention to have a further vote, probably a second reading of the withdrawal agreement bill before the European elections take place and hopefully in the much nearer future than that,” he said. “That is my understanding.”

Downing Street sources said it would be unlikely that the bill would be brought to parliament without being certain it could command a majority.

Members of the 1922 Committee have previously voted down proposals to change the party rules for a vote of no confidence in the Conservative leader, which currently mean they are protected for a year after winning a confidence vote, as May did in December 2018.

However, some on the executive have hinted that views could shift if May does not give a more explicit departure date, having previously said she would stand down when the withdrawal stage of the Brexit negotiations is completed.

Brady said there would be “further deliberations” following the meeting with the prime minister. “We will reach whatever conclusions we reach following that discussion,” he said.

Some May critics described the latest pledge as “yet more can-kicking” by the prime minister. Tory MP Nadine Dorries said: “She’s not given a decision yet. There’s no timetable, we need to get that decision soon because everyone needs it. That’s what I’ve asked Sir Graham for. The 1922 need to force her into that decision.”

Brady himself has pushed for May to consider bringing the withdrawal agreement bill to parliament, amended to include parts of the non-binding political declaration agreed with the EU, but tweaked to remove the controversial backstop plan for the Irish border.

Brady was behind a previous successful amendment which pledged to replace the backstop with “alternative arrangements” – an idea rejected by the EU after May returned for renegotiations in Brussels.

“I think that would be the best way forward but that is a matter for debate,” Brady said. “It remains the case that the only positive proposition that commands a majority is my amendment that sought to pass the withdrawal agreement without the backstop.

“I think bringing forward the second reading of the bill that sought to deal with the backstop is the best prospect of securing second reading. That is my view and goes beyond anything we’ve agreed.”

Bringing the EU withdrawal agreement bill to parliament without being certain of a majority is fraught with danger. Should the bill be voted down at second reading, it would not be possible to bring it back in this parliamentary session.

May could then be faced with the nuclear prospect of proroguing parliament and beginning a new session, including the mammoth task of passing a new Queen’s speech.

However, with little evidence of progress in the talks with Labour, and scant hope of securing agreement for indicative votes either, some cabinet ministers have begun to discuss a “plan C”.

It would involve laying out the concessions the government has made in the talks – and hoping sufficient Labour backbenchers could be won over to back the withdrawal agreement bill, even if Jeremy Corbyn had walked away.

Senior Tories have been maintaining the lines of communication that were opened when backbenchers including Oliver Letwin and Robert Halfon were working up a soft Brexit plan in concert with Labour backbenchers.

However, one Labour MP expressed scepticism about the idea, saying, “absolutely no chance – they wouldn’t get more than five to 10. Why would you break the whip and vote for it now, if you wouldn’t when we were a few days away from exit day and there was the imminent threat of no deal?”

How Theresa May's departure from No 10 could play out Read more

The government continued talks with Labour on Wednesday, despite deflation on the Labour side that May is prepared to offer anything substantial to secure support.

After the latest round of talks ended without agreement on Wednesday, a Downing Street spokesman said: “This is the second extended meeting between the govt and the opposition, which demonstrates the seriousness with which both sides are approaching these talks.” He added that there would be “further exchanges of documents as we seek to nail down the details”.

Labour struck a markedly more optimistic tone than on Tuesday, saying: “After the second day of talks this week, the negotiating teams are working to establish scope for agreement, and will meet again at the beginning of next week.”

Corbyn is set to launch Labour’s European election campaign on Thursday in Kent, with an appeal aimed at both leave and remain voters.

“Other parties appeal to just one side of the Brexit debate because they aren’t really committed to taking on the tax dodgers, the big polluters, or the financial gamblers who crashed our economy a decade ago,” he will say. “So whether you voted leave or remain in 2016, I urge you to vote Labour.”