Mr Tusk is expected to propose an alternative "flextension" plan to postpone the departure for a year while giving Britain the option of requesting an earlier exit if a deal can be agreed.
Downing Street said serious proposals had been made and it was prepared to change the wording of the deal to reach a compromise.
Instead, she was offered a short delay to April 12 - the date by which the UK must say whether it intends to take part in the European Parliamentary elections - or until May 22, if UK MPs had approved the withdrawal deal negotiated with the EU.
Ministers and Labour officials are holding a third day of talks created to establish whether they can unite around a compromise Brexit plan under which the Withdrawal Agreement would remain unchanged, but with changes being made to the accompanying Political Declaration on future relations.
The proposal by Mr Tusk has been met with scepticism from many, as Mr Tusk has a history of bad faith in his dealings with the British and has repeatedly expressed his desire for Britain to remain in the European Union, despite the Brexit vote.
The gamble ends up delaying Brexit and raising the possibility of closer future economic ties with the European Union, as May belatedly reaches out to opposition Labour MPs for help passing her agreement. If there was a lengthy extension to membership, the Tory party rulebook means she could not be forced out before December if she wished to go on.
A letter signed by the MPs, including shadow ministers, has been sent to Corbyn and members of the shadow cabinet and states that a public vote should be the "bottom line" in the negotiations.
It comes as talks continue between the Conservatives and Labour as the two parties attempt to thrash out a compromise deal.
Under the current terms agreed with the European Union, the United Kingdom is due to leave the bloc by May 22 if the British Prime Minister's repeatedly-rejected divorce bill clears the House of Commons or crash out without any deal in place by April 12.
And while German Chancellor Angela Merkel has said that "till the very last hour" Germany was determined to do everything possible to avoid a no-deal Brexit, others including French President Emmanuel Macron have signalled frustration.
Wednesday April 10: Mrs May will be in the Commons for Prime Minister's Question Time.
He has previously said he would want to hear a clear reason from May as to why Brexit should be delayed yet again - a move that he believes would add to uncertainties weighing on business across the bloc.
Finance Minister Bruno Le Maire told reporters in Bucharest: "If we are not able to understand the reason why the United Kingdom is asking for an extension, we can not give a positive answer".
Sterling fell against the USA dollar immediately after Labour's comments and stood 0.5 percent down on the day.
European Union leaders also reacted sceptically, saying that there had to be a strong justification for any further delay.
Justice Minister Barley tweeted that "there can only be an extension with a clear direction", but also proposed a potential way out by adding: "This includes the question of a second referendum".
They are also discussing the possibility of holding a confirmatory referendum.
Corbyn has been hard to pin down on whether he would insist on a new referendum, but his deputy Tom Watson said the party would have difficulty backing any deal without one. It is a prospect May and many in her Conservative party are anxious to avoid, fearing a backlash from voters.
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