To get agreement May will have to make major concessions on her red lines. She admitted there was a need to "break the logjam" and warned it was an "historic moment" for the nation.
Corbyn said British people need certainty that the country will not be "crashing out" of the EU without a deal. Labour will present May with its conditions for Brexit, which include a close economic relationship with the bloc, maintaining high environmental standards and protecting workers' rights.
May said she wanted the new plan in place in time for an emergency EU summit on April 10, so she can explain to EU leaders why Britain needs more time.
The EU has made clear any further extension will mean Britain taking part in EU elections on May 23. EU leaders must agree unanimously on a new delay and have warned they want a clear plan.
European Council President Donald Tusk urged the EU to be patient as it considers its response to May's offer to compromise on her divorce deal.
Tusk sent the tweet sent soon after May spoke at her Downing Street office.
Tusk's comment suggests that the EU will wait for Britain to present a clear plan. He tweeted: "Even if, after today, we don't know what the end result will be, let us be patient."
In a dramatic Downing Street speech, May said: "Today I'm taking action to break the logjam. I'm offering to sit down with the leader of the opposition and try to agree a plan that we would both stick to ensure we leave the EU and we do so with a deal.
"Any plan would have to agree the current Withdrawal Agreement - it has already been negotiated with the 27 other members and the EU has repeatedly said it cannot and will not be re-opened."
May said if she and Corbyn could not agree a way forward she would present "a number of options for the future relationship that we could put to the house in a series of votes to determine which course to pursue".
"Crucially, the Government stands ready to abide by the decision of the house. But to make this process work the opposition would need to agree to this too," she added.
Phones confiscated during marathon Cabinet meeting
Earlier, May and her warring Cabinet ministers were locked in Downing Street for eight hours as she held an emergency Brexit summit with just 10 days until Britain leaves the EU on April 12.
The Prime Minister is expected to address the nation shortly as tension mounts in Westminster over what the Cabinet might have decided. Ministers are still inside No 10 while Mrs May prepares the statement.
The Prime Minister's 'mother of all Cabinet meetings' massively overran with MPs having their phones confiscated to avoid leaks - and only allowed a sandwich and a stroll around the No 10 garden in a brief break from crisis talks.
May and her ministers have been hunkered down debating the way ahead for Brexit - and they entered Number 10 this morning bitterly divided between those who want Britain to leave the EU without a deal a week on Friday, and those who favour accepting a long delay and moving towards a 'softer' option.
They are trying to come up with a solution as Remainer MPs launch a new plot to stop a No Deal Brexit starting tomorrow - and following a warning from the country's top civil servant that crashing out of the EU could push food prices up by 10 per cent and plunge Britain into a recession.
Ministers finally brought their meetings to a close after 5pm - after gathering at 9.30am.
Fifteen of the Cabinet want to crash out without a deal, 10 back a softer Brexit and those on the losing side this evening could quit in disgust.
Ministers must decide on a way ahead before the rebel MPs led by Conservative Sir Oliver Letwin and Labour's Yvette Cooper force their hand by seizing control of Parliament again tomorrow
Letwin and Cooper will launch their plot to stop No Deal by passing a law compelling Theresa May to ask for a long delay to Brexit.
They had been hoping to pass a law to force a soft Brexit option on Theresa May, but after disastrous indicative votes last night ended in stalemate they will instead try to force through a bill compelling the PM to ask for a long delay at an EU summit on April 10.
- additional reporting AP
