Also on Wednesdy the European Central Bank blasted banks for slow-walking their Brexit preparations, telling them to move additional staff and resources to the European Union in case Britain leaves without a deal on October 31.
In response, Mr Corbyn plans to hold a no-confidence vote "at the earliest opportunity" and if successful will seek to form a caretaker government on a "strictly time-limited" basis.
Parliament has already voted several times - although non-bindingly - against a no-deal Brexit. He's got the support of his MPs and we are the biggest opposition party.
But the SNP's Ian Blackford welcomed the Labour leader's letter and Plaid Cymru's Liz Saville Roberts also offered her cautious support.
Sir Ed Davey, Lib Dem MP for Kingston and Surbiton who stood against Jo Swinson in the leadership election, spoke to Eddie Mair about the Labour leader.
She said, "Jeremy Corbyn is not the person who is going to be able to build an even temporary majority in the House of Commons for this task - I would expect there are people in his own party and indeed the necessary Conservative backbenchers who would be unwilling to support him".
The Labour leader has appealed to opposition leaders and Tory rebels to back his plan to bring down the Government with a no-confidence vote, extend the Brexit deadline beyond 31 October and force a snap general election.
Also on Friday, a source close to Dame Caroline dismissed any chance of her meeting with Mr Corbyn, despite the MP's name appearing on a letter suggesting she is open to talks.
It continued with Philip Hammond, the former Chancellor and prominent Remainer, threatening to vote against his own government in any motion of no confidence.
Welsh Conservative reaction to Mr Corbyn's letter was scornful, with Monmouth MP David TC Davies tweeting that the plan was like "a coup d'etat of the sort favoured by Marxist dictators in banana republics from whom he draws inspiration".
Lawmakers have two routes to stopping a no-deal Brexit - replacing Johnson's Government or passing a law that bans leaving the European Union without a deal.
Now their only policy that anyone in the country knows anything about is Brexit: stopping No Deal, which is obviously a national emergency at the moment, and they're not prepared to take action to stop No Deal from happening.
"Second, it means working together to make sure - whether it's before an election or afterwards - we solve this Brexit crisis with more democracy, not less democracy, by having a final say referendum so that all the people have their voice heard".