Prime Minister Gordon Brown has pledged not to "walk away", despite calls from former Lord Chancellor Lord Falconer for a debate about his leadership.
Speaking to Labour activists in London, the PM said he was "sticking" with people in their "time of need".
Lord Falconer had told the BBC he was "not sure" Labour could unite while Mr Brown remained leader.
Welsh Secretary Peter Hain has admitted that European election results, due out after 2100 BST, will be "terrible".
'Sticking with them'
Rallying Labour activists at a meeting in Newham, east London, Gordon Brown opened by joking that although his wife Sarah was not present "she hasn't resigned" - a reference to the six cabinet ministers who have left government in the past week.
He also joked that he had not realised Lord Mandelson had become "an expert in e-mailing and texting" - a nod to a story about a leaked e-mail in which the business secretary described the PM as "not comfortable in his own skin".
Mr Brown went on: "What would people think of a Labour government faced with an economic crisis... if ever we walked away from them at a time of need? We are sticking with them and working with them," he added.
He acknowledged it had been "a testing time" for the economy and politics but said: "What has been happening over these last few months is a test not just of our character, a test not just of the government, it is a test of our beliefs.
"If we believe that people should be responsible and people should act fairly and we should be fair to others, then it is our duty to make sure in our politics, in our economy, in our society, that's what happens."
BBC correspondent Ben Wright said Labour hoped to get the message across that ordinary party members wanted Mr Brown to stay.
But Lord Falconer, who was a close ally of former PM Tony Blair, told BBC One's Politics Show: "I think we are moving moderately quickly towards the need for a change and that change may be a change in leadership.
"We need unity above all. Can we get unity under the current leadership?
Labour probably needs a new leader to restore unity, says Lord Falconer
"I am not sure that we can and we need to debate it urgently and I think probably it will need a change in leader."
He denied being part of a "Blairite plot" against Mr Brown and said he admired the prime minister "greatly" but said he had an "inability to hold the party together".
The former Labour minister Nick Raynsford also warned that if there was no change the next general election could be "disastrous" and Labour reduced to "an ineffective and very small opposition party".
But the new Home Secretary Alan Johnson - the man widely touted as a possible replacement for Mr Brown - told the BBC he did not agree with Lord Falconer.
"I don't agree that regicide gives you a unified party," he said.
"I think Gordon Brown is the best man for the job."
Stop attacking PM, says Mandelson
He said Mr Brown was "a man of immense substance" who had "led the world" in dealing with economic problems.
"And I believe that he can do that job very very well and can lead us into election and can win that election."
Meanwhile, Cabinet Office Minister Tessa Jowell said that Mr Brown would be prepared to step aside if he became convinced that "he personally was an obstacle to Labour's recovery and achievement".
"Gordon Brown loves the Labour party. He is Labour through and through," she told BBC Radio 4's The World This Weekend.
Later aides insisted she was not suggesting Mr Brown should stand down but had been thinking of events in 2004 when some ministers persuaded Tony Blair to carry on at a time when he was considering quitting.
The next 24 hours are likely to prove critical for Mr Brown with the European results due to be declared from 2100 BST with the Labour leader set to address a meeting of all his MPs in Parliament on Monday at 1800 BST.
Conservative frontbencher William Hague said the prime minister and cabinet were "weakened" and Britain was "crying out" for an immediate general election.
Last week saw the departure of six cabinet ministers from the government and devastating local election results for the Labour Party, which saw it lose control of its four councils and its vote share drop to an historic low of 23%.
Peter Hain, who returned to the cabinet as Welsh Secretary in Friday's reshuffle, also predicted "terrible" European election results - due out Sunday evening.
Earlier, Business Secretary Lord Mandelson defended Mr Brown and said Labour could turn things around if it concentrated on policy and sorting out the expenses and economy.
He told Mr Brown's Labour critics: "Stop taking shots at the prime minister because you are simply going to make the situation for the party and government even worse."
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