Gordon Brown was facing public and private pressure to consider quitting for the sake of his party last night after the Crewe by-election “catastrophe” left ministers and Labour MPs convinced that they could not win with him at the helm.

Jack Straw, the Justice Secretary, was being earmarked by senior backbenchers as the figure to tell Mr Brown that they had lost confidence in him and that he should step aside unless there was a swift improvement in Labour’s fortunes.

Graham Stringer was the first Labour MP to call for Mr Brown to go, saying that the party needed a new leader to save it from “disaster” at the next election.

Ivan Lewis, the Health Minister, said that Crewe & Nantwich, where the Conservatives overturned a 7,000 Labour majority to win with their own majority of nearly 8,000, could mark the “beginning of the end” for Labour.

Cabinet sources said that a majority of ministers now doubted whether Labour could turn around its current deficit with Mr Brown in charge. Senior ministers were among those who told The Times privately that the party could not do nothing if it seemed to be heading towards certain defeat.

They said that a meeting of the Parliamentary Labour Party on June 2 would be critical to Mr Brown’s future. If the Prime Minister fails to impress, there is speculation that a senior party figure is ready to go public and call for him to quit.

One former Cabinet minister said there was a 20 to 30 per cent chance that Mr Brown would himself offer to step down if he believed that doing so would help his party.

No immediate move against Mr Brown appeared likely. Most ministers believe that he must be given until at least the conference season to re-establish himself. But last night Labour MPs were canvassing the names of candidates — Mr Straw, David Miliband, Alan Johnson, James Purnell and Andy Burnham — who they would expect to step forward if Mr Brown were persuaded to move out.

The Prime Minister brushed aside questions about his leadership and promised to steer the country through difficult times after the formerly safe Cheshire seat fell to the Conservatives with a 17.6 per cent swing.

David Cameron hailed the result as the “end of new Labour” and promised to build his own “coalition for change” to take the Tories back to power.

Mr Brown seemed highly unlikely to face a leadership challenge from former Cabinet ministers such as Charles Clarke and Alan Milburn, who have been deeply critical of him. MPs believe that such a move would scar Labour and mean certain defeat at the general election. But they said it was the role and duty of the Cabinet to tell Mr Brown that he had become a liability if they perceived that to be the general view in the party.

Mr Straw, as the most senior figure in the Cabinet, was being advanced as the ultimate “man in a grey suit” to tell the Prime Minister that he should consider quitting, a job he was unlikely to welcome because he would almost certainly regard himself as a potential leader to replace Mr Brown.

Cabinet ministers insisted there was no malevolence towards Mr Brown, and there there was a sadness for him that things had gone so wrong. But they said that Crewe had shown that people were no longer prepared to respect Mr Brown.

One senior critic of Mr Brown said the Crewe result was a catastrophe, particularly after the Government had suddenly come up with £2.7 billion to appease taxpayers — to no effect — in the middle of the campaign. That was the sum announced by Alistair Darling to compensate people who lost out from the 10p tax rate abolition.

The renewed burst of speculation about Mr Brown’s future dismayed No 10 and other ministers. Ed Balls, Mr Brown’s closest ally, went to his defence. The Schools Secretary said: “These are difficult times, times when you need strong leadership, values and experience. Gordon Brown has the right values and experience as a leader but it is not going to be turned around in a week.”

Tessa Jowell, the Olympics Minister, said: “If people think we are more interested in turning in on ourselves rather than facing up to dealing with the problems that people face in their daily lives, they will punish us for it.”

But Mr Stringer, a former whip, gave public voice to the private expressions of many ministers when he said: “Is it more damaging for the party to change the leader or cross our fingers and hope that things get better?” It was time, he said, for a senior Cabinet figure to mount a leadership bid to save the party from a “disaster” at the next election.

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