The shadow chancellor said "if anyone's to blame, it's me, full stop", but also cited Brexit and the media for having "demonised" the Labour leader ahead of the dismal defeat.
Mr Corbyn's key ally joined the leader in apologising on Sunday as the post-mortem examination was in full swing, with potential candidates to replace the leader setting out their stalls.
Key figures in the current leadership were tipping shadow business secretary Rebecca Long-Bailey, but backbenchers Lisa Nandy and Jess Phillips were testing the waters for a challenge.
Mr McDonnell followed the outgoing leader in apologising for losing dozens of seats across the North and the Midlands to the Tories on Thursday, which saw Labour's worst result since 1935.
"It's on me, let's take it on the chin, I own this disaster so I apologise to all those wonderful Labour MPs who have lost their seats and who worked so hard," Mr McDonnell told The Andrew Marr Show on the BBC.
But he also said he does "blame the media" which he said "demonised" Mr Corbyn "for four years solid, every day".
He predicted the leadership change will take place in eight to 10 weeks, tipping Ms Long-Bailey as having the ability to be "a brilliant leader".
Mr McDonnell also praised shadow cabinet ministers Angela Rayner, Dawn Butler and Richard Burgon, who himself was backing Ms Long-Bailey and said he is "considering" running as her deputy.
The shadow chancellor said he "prefers others" to Ms Phillips, before describing her as "really talented".
He said the next leader should be a woman - which would make her the first to lead the party - and said it was "most probably time for a non-metropolitan" candidate as he said "we need a northern voice".
Wigan MP Ms Nandy, a former shadow energy secretary, told Marr that she is "seriously thinking" about running for the leadership.
Conceding it is a "very hard road" to regain the trust of Labour voters in towns across the North, she called for the party's decision-making structures to move out of the capital.
"Our Labour headquarters, in my view, should move out of London, our regional offices should be empowered to take real decisions, we should move our party conferences back to towns as well as cities," she said.
Ms Nandy said it is "undoubtedly true" that Mr Corbyn is to blame for the devastating defeat, but said it was not a rejection of the ideas in the Labour manifesto.
Instead, she said: "We've got to rediscover how we can earn people's trust in order to make that radical change that the country needs."
Ms Phillips wrote a column in The Observer newspaper which was being seen as a potential pitch for a leadership challenge, though she is yet to throw her hat into the ring.
The MP for Birmingham Yardley, a Leave-backing constituency, said Labour was facing an "existential problem" that working-class voters do not believe the party is "better than the Tories".
"It's time to try something different," she wrote. "The truth is, there are corners of our party that have become too intolerant of challenge and debate."
Labour former minister Caroline Flint, who lost her seat in the former stronghold of Don Valley, said Mr Corbyn was not taking enough personal responsibility for the defeat.
The leader had written in an open letter that "I take my responsibility" for the loss and apologised, but had come under fire for an unrepentant tone on the night of the defeat.
"I will make no bones about it. The result was a body blow for everyone who so desperately needs real change in our country," he wrote to the Sunday Mirror.
Mr Corbyn said he will stand down in the early part of next year after overseeing a "process of reflection" within the party.
Ms Flint told Sky's Sophy Ridge on Sunday that his replacement cannot be "anybody who has had a hand in our Brexit strategy over the last few years".
"And I don't think it can be Corbyn without a beard, I think that would be the wrong move as well," she added.
Ms Flint said she was trying to "stem my anger" as she criticised "ardent Remainers" within the party - naming Sir Keir Starmer, Emily Thornberry, Hilary Benn and Yvette Cooper - for having "contributed to sacrificing 59 seats".
Labour former general secretary Iain McNicol called for Mr Corbyn to immediately resign so a caretaker Labour leader can hold the Tories to account.
Saying the opposition needs someone who can hold Prime Minister Boris Johnson's "feet to the fire" over Brexit, Lord McNicol told Ridge a party grandee like Harriet Harman, Mr Benn or Ms Cooper should be interim leader.