Theresa May is expected to meet Jeremy Corbyn later after she said she wanted to work with the Labour leader to break the Brexit deadlock.
The prime minister hopes the two of them can come up with a modified version of her deal with the EU that can secure the backing of MPs.
Mr Corbyn says he wants a customs union and workers’ rights to be priorities.
But Tory Brexiteer Boris Johnson has accused Mrs May of “entrusting the final handling of Brexit to Labour”.
Jacob Rees-Mogg, another prominent Brexiteer, described the offer as “deeply unsatisfactory” and accused Mrs May of planning to collaborate with “a known Marxist”.
Mrs May announced her plan to meet Mr Corbyn – as well as her intention to ask the EU for an extension to the Brexit deadline – after more than seven hours of talks with her cabinet on Tuesday.
BBC political editor Laura Kuenssberg says the latest move means the prime minister is likely to adopt a closer relationship with the EU – a softer Brexit – than she has agreed so far.
European Parliament Brexit co-ordinator Guy Verhofstadt, who had said he thought a no-deal Brexit was “nearly inevitable”, welcomed Mrs May’s offer of talks with Mr Corbyn.
“Good that PM Theresa may is looking for a cross-party compromise. Better late than never,” he tweeted.
Meanwhile, a cross-party group of MPs will attempt to push through legislation to stop a no-deal Brexit.
The UK was supposed to leave the EU on 29 March, but Mrs May agreed a short extension after MPs refused to endorse her withdrawal deal.
Attempts by MPs to find an alternative way out of the impasse also failed for the second time this week.
The UK now has until 12 April to propose a plan to the EU – which must be accepted by the bloc – or it will leave without a deal on that date.
Mrs May said she wanted to agree a new plan with Mr Corbyn and put it to a vote in the Commons before 10 April – when the EU will hold an emergency summit on Brexit.
If the two leaders do not agree a single way forward, Mrs May said a number of options would be put to MPs “to determine which course to pursue”.
In either event, she said she would ask the EU for a further short extension to the Brexit date to hopefully get an agreement passed by Parliament before 22 May so the UK does not have to take part in European elections.
The final decision on a delay rests with the EU. The BBC’s Europe editor Katya Adler said that while Brussels “welcomed” discussions between the PM and Mr Corbyn, its demands had not changed and it was “likely to put strict conditions on any further extension”.
Mr Corbyn said he was “very happy” to meet Mrs May and recognised his own “responsibility” to try to break the deadlock.
Labour has previously said it has six tests for judging any final Brexit deal, including protecting workers’ rights, establishing a permanent customs union with the EU and securing the same benefits of being in the single market the UK has currently.
She was for budging. The prime minister has made her priority leaving the EU with a deal, rather than the happy contentment of the Brexiteers in the Tory party.
For so long, Theresa May has been derided by her rivals, inside and outside, for cleaving to the idea that she can get the country and her party through this process intact.
But after her deal was defeated at the hands of Eurosceptics, in the words of one cabinet minister in the room during Tuesday’s marathon session, she tried delivering Brexit with Tory votes – Tory Brexiteers said “No”.
Now she’s going to try to deliver Brexit with Labour votes. In a way, it is as simple as that.
The offer of talks with the Labour leader provoked a backlash among Brexiteers, with Boris Johnson saying Brexit was “becoming soft to the point of disintegration”.
Jacob Rees-Mogg added: “To decide you’d rather be supported by a Marxist than by your own party is unwise.”
Andrea Jenkyns, a member of the pro-Brexit European Research Group, said Mrs May had failed to reach out to Conservative colleagues and she did not believe Mr Corbyn would work with the prime minister on a revised deal.
“He just wants chaos and wants to get into Number 10,” she told BBC Radio 4’s Today programme.
Asked what she would do in the event of a vote of no confidence in Mrs May’s leadership, Ms Jenkyns said: “That takes a lot of thinking about.”
Northern Ireland’s Democratic Unionist Party – which helps prop up Mrs May’s government but has repeatedly voted against her deal – said: “It remains to be seen if sub-contracting out the future of Brexit to Jeremy Corbyn, someone whom the Conservatives have demonised for four years, will end happily.”
However, Environment Secretary Michael Gove, himself a leading Brexiteer, said he backed the talks with Labour because he “wanted to ensure that minds are concentrated so that we do leave”.
After Mrs May’s statement, the president of the European Council, Donald Tusk, called for patience.
European leaders have been intensifying plans to cope with a possible no-deal, particularly surrounding the border between Northern Ireland and the Republic of Ireland.
If passed into law, the bill – presented by Labour MP Yvette Cooper – would require the PM to ask for an extension of Article 50 beyond that deadline.
Normally, the government chooses which bills to present to Parliament, but MPs previously voted to allow backbenchers to take charge of business in the Commons on Wednesday, meaning Ms Cooper can bring hers.
- Wednesday 3 April: Theresa May likely to begin talks with Jeremy Corbyn; cross-party group attempts to rule out no-deal in law
- Wednesday 10 April: Emergency summit of EU leaders to consider any UK request for further extension
- Friday 12 April: Brexit day, if UK does not seek / EU does not grant further delay
- 23-26 May: European Parliamentary elections