The Labour leader said regular televised debates would help the public “engage more in politics”.
SNP leader Nicola Sturgeon and the Lib Dems’ Tim Farron told The Independent they would take part – but No 10 said it wanted to see more details.
It comes amid press speculation Shadow Foreign Secretary Hillary Benn will be sacked next week in a Labour reshuffle.
Mr Corbyn is said to be keen for Labour to speak with one voice on defence and security issues, after the party’s divisions over whether to back bombing raids against IS targets in Syria.
But firing Mr Benn, who made an impassioned plea to Labour MPs to vote the opposite way to Mr Corbyn and back military action, could risk triggering further divisions.
Tensions were stoked by Labour whip Grahame Morris who urged Mr Corbyn to sack disloyal shadow ministers.
“This is what Jeremy Corbyn should do in 2016 – starting with a reshuffle that gets rid of mutineers,” he wrote on Twitter, in a post that was later deleted.
His comment provoked an angry response from Labour backbencher, and Corbyn-sceptic, Ian Austin, who said Mr Morris had previously been a serial rebel.
“Come on Grahame, name the ‘mutineers’!” he wrote on Twitter.
Asked if Mr Benn was facing the sack when MPs return from their Christmas break, Shadow Chancellor John McDonnell said: “Jeremy Corbyn is the leader of the party.
Mr Corbyn’s calls for an annual TV debate came after research by Leeds University found they increased viewers’ interest in politics by 30%.
He told The Independent: “It is crucial that the prime minister and government are held to account, both inside and outside Parliament, throughout their period in office – not just at election time.”
A senior Downing Street source dismissed Mr Corbyn’s challenge as a “desperate attempt by Labour to distract voters from the deep divisions that have left the party in turmoil”.
No 10 added that it had not been approached about the proposal and would need to see more details but a source said: “The PM is happy to be held to account every week at Prime Minister’s Questions by MPs.”
David Cameron pushed for the first televised election debates in 2010 – but proved reluctant to repeat the experience at this year’s election.
Tortuous negotiations with broadcasters resulted in a head-to-head debate with Labour leader Ed Miliband, a seven-way clash also including the Lib Dems, Green Party, UKIP, SNP and Plaid Cymru and one between opposition leaders.
It is thought any annual debate would have to adopt the wider format including all major parties.
Lib Dem leader Tim Farron accused Mr Cameron of being “two-faced” about TV debates.
“He should, for once in his life, match his words with deeds. I will lay out why the Liberal Democrats are the only real alternative to the Conservatives – both socially just and fiscally responsible,” he said.
A spokeswoman for the SNP said: “The televised debates earlier this year brought the election campaign alive and were far, far better for having a full range of participants – unlike in 2010 – which properly reflected the diverse range of political choices.”