Two possible candidates are the party's deputy leader Tom Watson and ex-shadow business secretary Angela Eagle.
But Mr Corbyn has said he would not "betray" his supporters by resigning.
He is expected to receive a warmer reception at a rally of his supporters on Wednesday.
Mr Corbyn was given a muted reception by his MPs during Prime Minister's Questions in the Commons, where David Cameron told him his presence as leader was "not in the national interest", adding: "For heaven's sake man, go."
On Tuesday 172 Labour MPs voted in favour of the no-confidence motion; 40 voted against.
Mr Corbyn's predecessor Ed Miliband and former deputy leader Harriet Harman have urged Mr Corbyn to step down, and Pat Glass has resigned as shadow education secretary - just 48 hours after being given the job after mass shadow cabinet resignations.
She said it was her "dream job" but added "the situation is untenable".
The former shadow chief secretary to the Treasury Seema Malhotra told Newsnight it was likely that a challenger would come forward on Wednesday.
Former acting leader Dame Margaret Beckett told BBC Radio 4's Today programme: "When you assume leadership you have to understand that the interests of those you lead come before your own and in those interests I'm afraid he should stand aside."
But shadow chancellor John McDonnell urged his fellow MPs to "play by the rules" and trigger a formal leadership contest.
He said Mr Corbyn was feeling "fine" but added: "We're all saddened that we're going through this because it's completely unnecessary."
Mr Corbyn has pointed to his backing among the party's grassroots, insisting that the vote by MPs had "no constitutional legitimacy".
"I was democratically elected leader of our party for a new kind of politics by 60% of Labour members and supporters, and I will not betray them by resigning," he said.
He will address a rally organised by his backers in the Momentum movement on Wednesday evening.
Also speaking in his support will be Public and Commercial Services union chief Mark Serwotka and Fire Brigades Union general secretary Matt Wrack.
Unite trade union general secretary Len McCluskey has accused the MPs voting for the no-confidence motion of "pointless posturing", warning they would have to mount a full-blown leadership challenge if they wanted to oust Labour's leader.
Labour MP Mike Gapes said Mr Corbyn's camp should "fear" Ms Eagle because she is "principled, competent and honest".
The result of Tuesday's no-confidence ballot has led to more calls for Mr Corbyn to make way.
Labour's leader in Scotland, Kezia Dugdale, suggested Mr Corbyn's position was untenable, telling the BBC: "If I had lost the support of 80% of my MSPs I could not do my job."
But following the result the leader issued a statement saying the government was "in disarray" following the vote to leave the EU, adding: "Labour has the responsibility to give a lead where the government will not."
He added: "Today's vote by MPs has no constitutional legitimacy.
"We are a democratic party, with a clear constitution. Our people need Labour Party members, trade unionists and MPs to unite behind my leadership at a critical time for our country."
The newly-appointed Shadow Energy Secretary Barry Gardiner also told Newsnight that MPs were using a back-door manoeuvre to undermine the elected leader.
"He has never had the confidence of the vast majority of Labour MPs. I didn't vote for him, I didn't nominate him.
"But the party chose in its wisdom to have him as its leader. We must work with that and then use the rules that the party has set to change the leader if we don't like him."
The new shadow cabinet line-up includes:
Shadow cabinet: Who’s in, who’s out?
On Monday, Mr Corbyn announced a reshaped shadow cabinet to replace those that had walked out but several positions in his top team remain to be filled after the mass resignations.
The shadow cabinet walkouts - in a bid to oust Mr Corbyn - came after the sacking at the weekend of shadow foreign secretary Hilary Benn, who told Mr Corbyn he had lost confidence in his leadership.
The SNP, meanwhile, is to ask the Speaker to declare it the official opposition at Westminster, claiming their Westminster leader Angus Robertson has more support than Mr Corbyn.
Labour MPs are expected to announce a challenge to Jeremy Corbyn's leadership after a motion of no confidence in him was supported by three-quarters of the parliamentary party.