Theresa May is seeking an extension to the new Brexit date of April 12, in order to negotiate a way to "break the logjam" with opposition leader Jeremy Corbyn.
Mr Corbyn said he would be "very happy" to meet the Prime Minister in a bid to offer "certainty and security" to the British people - but Conservative Brexiteers have reacted with anger.
The move comes after Mrs May chaired a marathon Cabinet meeting yesterday for Brexit 'crisis talks'.
A source said ministers at the meeting were split 14-10 against asking for a long extension to the Brexit process.
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."
Jacob Rees-Mogg described the offer as "deeply unsatisfactory" and accused Mrs May of planning to collaborate with "a known Marxist".
However, European Parliament Brexit coordinator 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.
The April 12 Brexit deadline was set if Theresa May's withdrawal agreement failed to gain support in the Commons.
If the EU grants an extension beyond May 22, it is understood it would be possible for the UK to prepare for the European Parliament elections on May 23, but then cancel them at the last minute if the withdrawal deal was ratified.
On Wednesday, the EU will meet in Brussels for the first of a series of briefings on Brexit preparedness, and European Commission president Jean-Claude Juncker will speak about the UK's withdrawal.