Theresa May should set a date next week for her departure, the chair of the Tory backbench 1922 Committee has said.

Sir Graham Brady said it would be strange if a scheduled meeting with the prime minister and the group’s leaders next week did not result in a clear understanding of her departure timetable.

May indicated earlier this year that she would resign once her Brexit deal with Brussels has been passed by MPs – a promise that seemed to persuade a number of Brexit-backing Tory MPs to back the deal, among them Boris Johnson and Jacob Rees-Mogg.

However, parliament has been in deadlock over Brexit for months and has rejected the deal three times.

Brady, the MP for Altrincham and Sale West, told BBC Radio 4’s Week in Westminster programme: “I don’t think it’s about an intention for staying indefinitely as prime minister or leader of the Conservative party. I think the reticence is the concern that by promising to go on a certain timetable, it might make it less likely she would secure parliamentary approval for the withdrawal agreement, rather than more likely.”

At this week’s PMQs, Andrea Jenkyns, a pro-Brexit Tory MP, asked May whether it was time for her to “step aside and let someone else lead our country, our party and the Brexit negotiations”.

May rejected the call and reiterated her belief that it was not her fault that the Brexit process had ground to a near standstill. “If it were an issue about me and the way I vote, we would already have left the European Union,” she said.

Downing Street said May would not make any further concessions beyond her original resignation offer to the 1922 Committee.

“The PM made a very generous and bold offer to the 1922 Committee a few weeks ago that she would see through phase one of the Brexit process and she would leave and open up for new leadership for phase two,” a No 10 source said.

May’s decision to work with Labour on a compromise to deliver Brexit seems to have caused many Tories to lose their patience. Tensions within both the Conservatives and the Labour party have grown and cost both parties dearly at local elections this month, and could lead to similar losses in the European elections on 23 May.

One of the Conservative party’s prominent financial backers, Jeremy Hosking, told the Financial Times he had donated £200,000 to Nigel Farage’s Brexit party because the European elections had become, by default, a second referendum on Brexit.