The leaders of the EU's 27 remaining countries gave the United Kingdom until April 12 to leave the bloc or to come up with a new plan, after lawmakers thrice rejected an agreement struck between the bloc and May.
Al Jazeera's Neave Barker - reporting from outside 10 Downing Street, May's official residence - said senior ministers had been in talks for nearly seven hours.
The minister said if Parliament were to back an agreement later this week, it would be possible to avoid a long extension to EU membership, and being forced to take part in European Parliament elections next month.
The EU's chief Brexit negotiator said this morning: "A no deal was never our desired or intended scenario, it was never my desire or intended scenario".
British Prime Minister Theresa May will be chairing a crucial meeting of senior ministers on Tuesday to seek a way out of the Brexit deadlock as the European Union warns a no-deal departure is growing more likely by the day.
Speaking at the Élysée Palace alongside Irish Prime Minister Leo Varadkar, Macron said it was for Britain to decide whether the plan involved new elections, a referendum or a customs union. "And we'll ensure that those are on the table", he said. "Being prepared for no deal does not mean that there will be no disruption", he said.
Theresa May made her announcement after a marathon cabinet meeting that lasted most of the day.
"We are now in a really unsafe situation with a serious and growing risk of no-deal in 10 days' time", opposition Labour MP Yvette Cooper, who presented the bill, said on Tuesday.
The call for a unified approach followed a nine-hour Cabinet session as senior ministers hammered out a road map.
Second referendum: The proposal for a confirmatory referendum on May's deal earned the highest number of affirmative votes, though it was defeated 268-295.
The prime minister's offer to Corbyn is another sign of the desperation and disarray that has overtaken her government as it struggles to complete Britain's acrimonious separation from the EU.
NCAA Tournament roundup: Okeke injured as Auburn upsets UNC
The No. 2-seeded Kentucky Wildcats held on against the No. 3- seeded Houston Cougars, 62-58, in a Friday night Sweet 16 battle. As Okeke left the court, players from North Carolina came over and wished him well in a show of good sportsmanship.
But, the source added, there may be more "nuance" between their positions on whether to give May more time when she next meets them on Wednesday next week.
Among the possibilities left in Mrs May's hands, there is calling for a snap election, holding a fourth meaningful vote in Parliament on her withdrawal agreement, asking Brussels for a long extension or leaving the European Union without a deal on April 12.
May will put her deal back to Parliament again, but only if there's a chance of success, he said.
If passed into law, the bill would require the PM to ask for an extension of Article 50 - which mandates the UK's exit from the European Union - beyond the current 12 April deadline.
Labour legislator Yvette Cooper, one of the bill's sponsors, says "we are now in a really risky situation with a serious and growing risk of no-deal in 10 days' time".
"The deal is a deal which can appeal both to people who voted leave and to people who voted remain", he told the BBC.
Parliament on Monday rejected four alternatives, with proposals for a customs union failing by three votes.
Despite the downbeat assessment, Barnier did say that "we can still hope to avoid it" through intensive work in London ahead of an April 10 summit.
A long list of global corporations have already announced plans to relocate their European headquarters from London to other cities in the EU over Brexit, and others have already shifted some personnel and put contingency plans in place to move more out of the Britain.
But Brexit Secretary Steve Barclay indicated the government's immediate plan was to have one more go at getting May's unpopular deal approved.