It is believed the former shadow business secretary has the support of the 51 MPs needed to mount a challenge.
Ms Eagle was one of 20 members of Mr Corbyn’s shadow cabinet to quit since Sunday, which led to MPs passing a motion of no confidence in him.
Mr Corbyn’s allies say they believe he will win an election and have urged it to take place as quickly as possible.
Labour Deputy leader Tom Watson says he has tried to persuade Mr Corbyn to stand down, but has ruled out a leadership bid himself.
Trade unions indicated that they would not seek to stand in the way of a contest provided it was carried out “through the proper democratic procedures provided for in the party’s constitution”.
In a speech to student supporters in London last night, Mr Corbyn said while he recognised not everybody supported the direction he was taking the party, he had the backing of party members to carry on.
“I also recognise that the mandate was given by hundreds of thousands of ordinary people joining in the political process,” he said to cheers from supporters.
Mr Corbyn faced another humiliating day yesterday as even Prime Minister David Cameron took the unusual step of calling on him to go in the “national interest” in front of MPs in the Commons.
“For heavens sake man, go!” Mr Cameron said to cheers from Tory MPs.
Mr Corbyn’s predecessor Ed Miliband also joined the chorus of calls for him to stand down, insisting his position is now “untenable.”
Ms Eagle is now the frontrunner to stand as a “unity” candidate but may run into problems over her support for the Iraq war with the Chilcott inquiry into the conflict being published next week.
The civil war in the party has also spilled over into Scotland as a group of activists backing Mr Corbyn attacked rebel MPs, while more than 200 members, including MSPs and ex-Parliamentarians came forward calling on the leader to quit.
Mr Corbyn overwhelmingly lost a vote of confidence in his leadership among MPs by 172 votes to 40. But any leadership contest would be decided by grassroots members and he was elected nine months ago with overwhelming support around the country.
Mr Watson said: “He’s obviously been told to stay by John McDonnell and his team, and they’ve decided they’re going to tough this out. It looks like the Labour party is heading for some kind of contested election.”
But the shadow chancellor said it was “ludicrous” to suggest he was trying to prevent Mr Corbyn from quitting.
He said: “Having just seen Jeremy, he is buoyant, he is campaigning, he is doing his work here. He is not going anywhere.”
Mr Miliband said he had been supportive of Mr Corbyn but is now urging the Opposition leader to now reflect on what is “the right thing for the country”.
Britain is facing its worst crisis since the Second World War, Mr Miliband said.
Daniel Johnson, MSP for Edinburgh Southern, Catherine Stihler MEP and former shadow Scottish secretary Margaret Curran are among the Labour members in Scotland who have signed a letter calling on Mr Corbyn to quit.
It followed an earlier open letter to Edinburgh South MP Ian Murray from 80 Scottish Labour figures, including Elaine Smith MSP, condemning him for handing a “gift” to the SNP and Tories when he quit as Shadow Scottish Secretary earlier this week.
There was even speculation last night that Scottish leader Kezia Dugdale had offered to serve as shadow Scottish Secretary rather than have a non-Scottish MP fill the role in the shadow cabinet.
Mr Corbyn’s spokesman issued a defiant message to rebel MPs yesterday.
He said: “They have a simple choice as we see it: they either unite behind the leadership of the democratically-elected leader or they trigger a leadership election.”
“We are calling on Labour MPs to recognise that they were elected to represent the Labour Party as the Opposition. If they don’t want to do that, they should trigger a leadership election.”