The prime minister confirmed the UK would seek a further short delay to Brexit beyond 12 April in a Downing Street statement that infuriated Eurosceptic Tories.
Government insiders believe the two big parties’ positions on future customs arrangements are not so far apart and could form the basis of a new deal.
No 10 aides said the PM would approach talks in a “constructive spirit”, but within minutes other sources suggested discussions would simply be used to recapture control of the Brexit agenda as MPs come close to forcing an outcome on the government.
Mr Corbyn agreed to a meeting, possibly set to take place as early as Wednesday, saying he recognised his responsibility “to represent the people that supported Labour” at the last election.
Ms May’s move came after a marathon seven-hour cabinet during which ministers had mobiles taken off them to prevent clandestine briefing to the press ahead of the prime minister’s statement.
Brexiteer ministers who have said they would quit if the PM pursued a customs union were yet to resign last night with the final Brexit outcome still unclear.
In her statement Ms May said: “Today I am taking action to break the logjam.
“I am offering to sit down with the leader of the opposition and 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.”
The prime minister said the idea would be to agree on a joint approach both she and Mr Corbyn can back and if not, put forward several different options for future relations with the EU for the Commons to vote on.
The Independent understands Ms May would be likely to bring back her own previously defeated deal as one of the options at that point, if initial talks with Mr Corbyn fail.
The prime minister said she would “abide by the decision” of the Commons and that Mr Corbyn would have to as well for the approach to work.
The Labour leader responded: “We recognise that she has made a move.
“I recognise my responsibility to represent the people that supported Labour in the last election and the people who didn’t support Labour, but nevertheless want certainty and security for their own future and that’s the basis on which we will meet her and we will have those discussions.”
The PM’s aides would not rule out that discussions could include a customs union or even giving the British public a Final Say referendum on Brexit, although they pointed to the PM’s previously stated views.
The government’s current plan for future relations include many facets of a customs unions, while Mr Corbyn’s approach calls for “a customs union” but one that would give the UK greater freedoms than Britain’s existing EU membership.
The Conservative Party is already in turmoil, with one MP quitting on Monday claiming the cabinet is “cowardly and selfish” – but the idea that Ms May could strike a deal with Labour will probably push the Tories to the brink of a major rupture.
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.”
He said it would be “ridiculous” if the UK were forced to contest European elections as a result of the further delay, adding: “I can under no circumstances vote for a deal involving a customs union as I believe that does not deliver on the referendum.”
Ms May delivered her statement while the Brexit-backing European Research Group (ERG) met, with its members watching her on television and one reporting, “it did not go down well”.
Chair of the ERG Jacob Rees-Mogg attacked Ms May for working with “a known Marxist”, and when asked if he could back a deal struck with Mr Corbyn, he said: “Well, I wouldn’t hold your breath – I’m not a Marxist in case you hadn’t noticed.”
He added: “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.”
But one minister told The Independent that the move was about ensuring that a cross-party move by MPs, led by ex-Tory minister Sir Oliver Letwin and Labour’s Yvette Cooper, to impose an outcome on the government is superseded, with the Commons looking to vote on its own options again soon.
Cabinet ministers were called to Downing street at 9.30am on Tuesday for a stormy meeting, one without civil servants, to discuss the way ahead.
It lasted until 2pm, with ministers eating sandwiches for lunch, before going into a formal government cabinet until 5.30pm.
Ms May’s aides were clear that there was little support for a general election, but would not confirm there had been unanimous backing for her suggested approach, with The Independent hearing differing reports of the level of opposition.
One suggested that the majority for an extension in cabinet was 17 to four, with only two of the 17 calling for a long one and two further making a point about backing a short one.
Another very different account suggested that there were 14 ministers against the idea of an extension and only 10 for. Ms May’s aides later said there had not been a specific show of hands at the meeting.
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