The shadow education secretary, Angela Rayner, has cleared the way for her friend Rebecca Long-Bailey to run as the leftwing successor to Jeremy Corbyn, deciding instead to focus on becoming deputy leader.

After Labour’s catastrophic performance in last week’s general election, Corbyn said he would remain in place until March, while a leadership contest takes place.

Long-Bailey, the shadow business secretary, had long been regarded as the leadership’s pick for the top job. She is expected to win the backing of the party’s largest donor, Unite, and of key Corbyn allies including elections chief Karie Murphy.

Long-Bailey, who is the MP for Salford, was given a prominent role during the election campaign, including as Labour’s spokeswoman in TV debates, and has stood in for Corbyn at PMQs. But Rayner has built up a significant public profile, and was regarded as a strong potential leadership contender.

However, in a move one colleague described as “sisterly” she is now expected to run for deputy leader instead – smoothing the path for Long-Bailey.

Allies of Rayner said she had yet not made a final decision but that discussions with colleagues were focused on the deputy position.

Other Labour MPs expected to run for the leadership include shadow Brexit secretary Keir Starmer, Wigan MP Lisa Nandy, shadow foreign secretary Emily Thornberry and high-profile backbencher Jess Phillips.

The leadership race, which will be held during the next three months, will determine whether Labour maintains the leftwing policy platform espoused by Corbyn and his supporters – or abandons some of these elements.

Phillips said on Sunday that Labour must “discover the courage to ask the difficult questions about the future of our party and the future of the working-class communities who need a Labour government”.

Starmer and Thornberry have been blamed by some on the left of the party, including Unite’s general secretary Len McCluskey, for dragging Labour towards supporting a second Brexit referendum, which they claim is what lost the party support in scores of leave-voting seats.

McCluskey claimed last week’s defeat resulted from, “a slow-motion collapse into the arms of the People’s Vote movement”.

However, key Labour figures including shadow chancellor John McDonnell believe that without a shift towards remain, the party’s activists would desert it, and anti-Brexit voters would switch to the Lib Dems.

The post of deputy leader was vacated by Tom Watson at the outset of the general election campaign, when he opted not to run as a Labour candidate. His West Bromwich East seat was one of those Labour lost to the Conservatives.

Rayner, 39, a former care worker and union rep, attacked Boris Johnson during the campaign, after a 1995 article emerged in which he condemned single mothers as raising a generation of “ill-raised, ignorant, aggressive and illegitimate children”.

She said: “People like him make women who are already vulnerable feel that they’re the problem. They’re not the problem.”

Describing how she felt when she had her own first child at 16, she said: “I didn’t want to feel ashamed. I wanted to be the best mum I could be. I just wanted the means to be able to help myself.”

She is likely to face candidates including Dawn Butler and Richard Burgon in the contest to be deputy leader.

Friends of Burgon say he believes Labour is a “broad church”, and the race for deputy leader should include a candidate from the leftwing Campaign Group of MPs, of which he is secretary.

Recriminations continue to fly after the election result, with Thornberry threatening to sue her former colleague Caroline Flint.

Thornberry said she had consulted lawyers after Flint, who lost her Don Valley seat last week, claimed on live television that Thornberry had once said to a fellow MP in a leave-voting seat that she was “glad my constituents aren’t as stupid as yours”.

Thornberry told Sky News: “I’ve contacted her and I’ve said to her: ‘Please withdraw, I’ll give you until the end of the day.’ And she hasn’t. So I’ve had to go to solicitors. People can slag me off as long as it’s true, I can take it on the chin. But they can’t make up shit about me – and if they do, I have to take it to the courts.”

She added: “It’s ridiculous, absolutely ridiculous. I have better things to think about than people going on television and making up shit.”

Flint made the comments in an interview with Sky on Sunday, in which she claimed that the next Labour leader should be someone who had not pushed the party towards a pro-remain position, as Thornberry and Keir Starmer had.

Sadiq Khan joined the criticism of Corbyn’s handling of the election and its aftermath on Monday.

The mayor of London wrote in the i paper: “If we are truly honest with ourselves, Labour simply did not put forward a credible candidate for prime minister or a believable set of priorities for governing. All the evidence – not least from the doorstep and the results – backs this up.”

In an apparent criticism of Corbyn’s article in Sunday’s Observer, in which the Labour leader claimed his party had “won the arguments and rewritten the terms of political debate,” Khan said, “it is ludicrous to say we won the argument at this election – because we lost. Shifting the dial on the debate around austerity is no consolation to those most affected by Tory rule.”

Labour’s general secretary, Jennie Formby, wrote to Labour’s ruling national executive committee on Sunday to say a full timetable for the leadership contest would be agreed, with a recommended start date of 7 January.