Gordon Brown has said he can steer Britain through "difficult" economic times - amid signs of growing backbench disquiet over his leadership.
Mr Brown reacted to Thursday's crushing by-election defeat in Crewe and Nantwich by stressing his economic record and pledging to listen.
But Labour MP Graham Stringer said the party needed a new leader to save it from "disaster" at the next election.
And Alan Simpson said he had until the end of the year to "turn things round".
The Cabinet has been rallying round Mr Brown after the Tories' historic by-election victory - and brushing off suggestions that he is not the right person to lead Labour into the next general election.
Labour deputy leader Harriet Harman said the vast majority of Labour members were "solidly behind" Mr Brown.
But BBC political editor Nick Robinson said privately some ministers had expressed concerns about Mr Brown's leadership - one minister had compared the party's plight with that under Michael Foot's leadership.
It is the responsibility of senior members of the Cabinet to say we are going in the wrong direction
Graham Stringer Labour MP
Graham Stringer earlier became the first Labour MP to call for Mr Brown to go, telling the BBC News Channel: "Is it more damaging for the party to change the leader or cross our fingers and hope that things get better?".
He said it was time for a senior Cabinet figure to mount a leadership bid to save the party from a "disaster" at the next election.
"I think if the party is to renew itself, get its policies in line with what the people we represent want, then it's the responsibility of senior members of the cabinet to say we're going in the wrong direction, it's impossible to change the situation we're in at the moment, and to say to Gordon that they intend to stand for election.
"Without that we're heading for an electoral disaster at the next general election and I desperately want the Labour party to win."
Fellow backbencher Alan Simpson said voters were fed up with "cheap politics" and warned Mr Brown had until the end of this year to turn things around.
"We have let voters down on some totemic issues," he said, naming post office closures, police pay and 10p tax.
He said Mr Brown had "until the end of the year to change direction ...or I think it's the end of the pier".
Conservative leader David Cameron said Labour had run a "backward looking" and "divisive" by-election campaign: "It was in many ways the end of New Labour and I think it was a great mistake for them."
But Ms Harman said the result had not been "a judgment from voters about the campaign" or a judgment on the candidates - instead it was about "family finances and apprehension they fear about their prospects" which the government would address.
She also said by-elections were not "good predictions of what's going to happen at a general election" and Labour had taken a "bigger hit" in several by-elections before the 2005 general election - but gone on to win.
Labour strategists were insisting the by-election had been a "referendum" on the abolition of the 10p tax band and voters' concerns over rising prices - not Mr Brown's leadership.
Speaking on a visit to Guy's and St Thomas' hospital, Mr Brown said he had got a "clear and unequivocal" message from voters in Crewe and Nantwich.
"The message that we have got is that people are concerned. They're concerned about rising food prices, rising petrol prices. People are concerned, rightly, about gas and electricity bills," he said.
"And I think the message that I have to get to people is this, that we are unequivocal and clear in our direction, that we're going to address and are addressing these problems. We will continue to do so.
"And my task is to steer the British economy through what have been very difficult times in every country of the world."
He was asked whether nervous Labour backbenchers might try to move against him, but replied the "task ahead" was to take the British economy through "difficult times".
On Thursday the Conservatives made their first by-election gain since 1982, winning the seat from Labour on a 17.6% swing.
Labour left-winger John McDonnell, who failed to get enough Labour MPs backing him to challenge Mr Brown for the leadership last year, said Mr Brown's "re-launch" after the local elections had been a "disaster".
"Things are just going from bad to worse for the government," he said.
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