May said: "I am taking action to break the logjam. I am offering to sit down with the Leader of the Opposition and to try to agree a plan - that we would both stick to - to ensure that we leave the European Union and that we do so with a deal."
Brexiteers fear May is about to pivot to supporting a customs union, despite only 37 Tory MPs voting in favour of one when MPs considered alternative Brexit outcomes yesterday.
If May and Corbyn fail to agree a way forward before next week's emergency EU summit the Government will give MPs another chance to decide on an alternative and would "abide by" the decision of the House of Commons.
Jacob Rees-Mogg, the leader of the European Research Group of Tory Brexiteers, said: "What was announced today was an attempt to overturn the referendum that wanted a clear Brexit, to do a deal with a socialist who doesn't want Brexit and is not in line with the Conservative Party's manifesto commitments or with the referendum.
"This is all about getting further and further away from the Brexit people voted for, which is very serious."
Former foreign secretary Boris Johnson said: "It is very disappointing that the Cabinet has decided to entrust the final handling of Brexit to Jeremy Corbyn and the Labour Party. It now seems all too likely that British trade policy and key law making powers will be handed over to Brussels - with no say for the UK.
"As it is, we now face the ridiculous possibility of being forced to contest the European elections more than three years after leaving the EU and having to agree to exit terms that in no way resemble what the people were promised when they voted to leave."
Johnson added: "If Jeremy Corbyn gets his way we remain in the customs union. We should be leaving. You could manage a perfectly successful no-deal outcome. That is what the public wants."
Eurosceptic ministers emerged despondent from the Cabinet meeting and predicted that members of the Cabinet could now quit because they cannot back May's plan.
The 14 ministers who spoke out against an Article 50 extension were Liam Fox, Gavin Williamson, Penny Mordaunt, Liz Truss, Sajid Javid, Chris Grayling, Jeremy Wright, Andrea Leadsom, Jeremy Hunt, Baroness Evans, Steve Barclay, James Brokenshire, Alun Cairns and Brandon Lewis.
Those who spoke in favour of a long extension were Philip Hammond, Matt Hancock, David Gauke, Greg Clark, David Lidington, Karen Bradley, Michael Gove, Amber Rudd, Claire Perry and Geoffrey Cox.
During the marathon Cabinet meeting Hammond, the Chancellor of the Exchequer, suggested the option of a "confirmatory vote" on any Brexit deal - in other words a second referendum - should remain on the table, according to several ministers. Hammond denies backing another referendum.
In a televised address, May said: "I have always been clear that we could make a success of no-deal in the long-term, but leaving with a deal is the best solution.
"So we will need a further extension of Article 50 – one that is as short as possible and which ends when we pass a deal."
Corbyn, who had no advance warning of the offer, said he would meet May because "I recognise my responsibility" but would not say whether he was prepared to compromise on his red lines, which include the need for a customs union with the EU.
Downing Street insisted that Britain could still leave the EU on May 22 if a compromise was agreed, meaning the UK would not need to take part in the European Parliament elections.
