A small group of Labour MPs look set to announce their resignations from the party, senior party sources have said.
Several backbenchers, who disagree with the Labour leadership over its handling of Brexit and the anti-Semitism row, are expected to break away.
But Labour MP Stephen Kinnock, a vocal critic of leader Jeremy Corbyn, has urged them to “stay in and fight”.
Shadow chancellor John McDonnell said on Sunday he did not see “any need for anybody to split from the party”.
“Those saying we’ll split if we don’t get a ‘People’s Vote’ [another referendum on the final Brexit deal] – well, we’ve still kept that option on the table and it might come about,” he said.
Mr McDonnell insisted the party was “holding together on Brexit” and would be “ruthless” on claims of anti-Semitism that have plagued Labour.
He said the effect of MPs splintering from the party would be similar to the SDP breakaway in 1981, which split Labour’s vote and “installed Mrs Thatcher in power for a decade”.
BBC political correspondent Iain Watson said senior Labour sources had told colleagues to expect four resignations on Monday morning.
None of those believed to be considering resigning were responding to requests for comment on Sunday night.
Mr Kinnock, Labour MP for Aberavon, told the Westminster Hour on BBC Radio 4 there would “probably be some kind of splintering”.
He continued: “We have a duty to deliver for people that we are elected to represent and to change the country for the good and the Labour Party is the vehicle to do that.”
Asked what his message was for MPs thinking of leaving the party, he said: “I would say to them that there are certain people in the party with whom they disagree very strongly.
“The way to win the battle, if you like, is to stay in and fight.”
A former Labour vice-chairman said on Sunday he would quit the party over what he saw as a repeated failure to tackle hostility towards Jewish people.
Former Barnsley East MP Michael Dugher, who stood down at the last election, told the Sun: “I can no longer justify paying subs to a party which I now regard as institutionally anti-Semitic.”
BBC political correspondent Ben Wright said speculation that several Labour MPs could be poised to break away from the party is mounting.
“The leadership’s handling of Brexit and anti-Semitism in particular has fuelled the disillusionment of some Labour MPs,” our correspondent said.
Last week, Labour shadow Treasury spokesman Clive Lewis warned of “severe” ramifications if the party was seen to facilitate a “Tory Brexit”.
Meanwhile, ex-shadow chancellor Chris Leslie said it was “heartbreaking” that Labour was not united in arguing against leaving the EU.
On Sunday, Labour members on social media began circulating a graphic pledging to work towards a Labour-led government “under whatever leadership members elect”.