Labour MPs have called for a motion of no confidence in Jeremy Corbyn's leadership.
The move has come from Labour MPs Margaret Hodge and Ann Coffey.
Mr Corbyn has been accused of running a half-hearted EU campaign that failed to voters in Labour's heartlands.
Another Labour MP Angela Smith has called on Mr Corbyn to "consider his position".
Ms Smith accused the Islington MP of showing "insufficient leadership".
Former shadow chancellor Chris Leslie said he was "not surprised" at the move and called on Mr Corbyn to consider his position.
"I would say today he does need to consider his position and think about whether he should do the honourable thing," he told BBC News.
"Every MP is going to have to search their conscience about it but I think I would need an awful lot of persuading to have confidence in Jeremy's leadership going into a general election."
Dame Margaret urged Mr Corbyn to "do the decent thing" and stand down in the way that David Cameron had.
"The European referendum was a test of leadership and I think Jeremy failed that test," she told Sky News.
"He came out too slowly, he was very half-hearted about his attempts to campaign and Labour voters simply didn't get the message."
Dame Margaret said any vote in the Parliamentary Labour Party would be by secret ballot so rebels would not have to declare their opposition publicly.
A spokesman for Mr Corbyn condemned the move as "self-indulgent" when the UK faced real problems in the aftermath of the Brexit vote.
He acknowledged, however, that it had been discussed at a post-referendum meeting of the shadow cabinet.
Mr Corbyn has faced criticism for failing to voters in Labour's traditional heartlands.
Many labour MPs believe those voters are turned off by a veteran socialist who sits for a north London seat.
Lord Mandelson, the Labour cabinet minister and an ex-EU commissioner, said that the campaign had shown that Mr Corbyn could not "cut it" as leader.
Senior Labour Leave campaigner Gisela Stuart also called on him to consider on his position.
"It is now incumbent to all party leaders to also reflect to what extent they are representing and reflecting the views of the voters," she said.
And anger that has been simmering about Mr Corbyn for months spilled over.
Frontbencher Chris Bryant hit out at former leader Ed Miliband, whose overhaul of the party's rule book in the wake of allegations of vote rigging in Falkirk paved the way for Mr Corbyn's election as leader.
"I might go and punch him because he's a tosspot and he left the party in the state it's in," Mr Bryant said.
Mr Corbyn, a long-time Eurosceptic, insisted even before the result that he would take no blame for a Brexit.
He blamed the outcome on the Conservative Government's austerity cuts, which he said had alienated voters.
"A lot of the message that has come back from this is that many communities are fed up with cuts, they are fed up with economic dislocation and feel very angry at the way they have been betrayed and marginalised by successive governments in very poor areas of the country," he said.
"The point I was making was there were good things that had come from Europe in working conditions and environmental protections but there were other issues that had not been addressed properly."
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