"We want to leave the European Union on April 12 or very soon afterwards", The Sun quoted a Brexit-supporting minister as saying in reference to the letter.
Britain's Prime Minister Theresa May speaks in the Parliament in London, Britain, March 29, 2019 in this screen grab taken from video.
It follows defeats by even wider margins in January and March, and leaves the government's blueprint for exiting the bloc in tatters.
Britain has in the past had national unity governments in times of national crisis, such as World War I and II, but critics have been quick to point out such coalitions were forged when there was a single goal, like defeating Nazi Germany.
"Some Number 10 advisers are pushing for the run-off (between May's Brexit deal and alternatives) to come as early as Tuesday, meaning an election could conceivably be called as early as Wednesday", The Mail on Sunday, without citing sources.
"There are no ideal choices available and there are very good arguments against any possible outcome at the moment but we are going to have to do something", said Justice Secretary David Gauke, who voted in the 2016 referendum to stay in the EU.
The vote was held on the day the United Kingdom had initially been scheduled to leave the EU.
On Friday, Members of Parliament voted 286 to 344 against the agreement, leaving the country's future more uncertain than ever. In a first round, the idea of seeking a customs union with the European Union was one of the most popular options, although it fell just short of a majority.
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On Monday, Parliament will hold an indicative vote on Brexit alternatives.
But others refused, including May's Northern Irish allies, the Democratic Unionist Party (DUP), which says planned arrangements for the Irish border after Brexit are unacceptable.
"Since it took two and three-quarter years for the Government to get what it had negotiated defeated three times, it's a little bit harsh on Parliament, when it started the process last Wednesday, for not having immediately solved the problem in 24 hours", Mr Benn said.
"In view of the rejection of the Withdrawal Agreement by the House of Commons, I have made a decision to call a European Council on 10 April", said European Council President Donald Tusk, immediately responding to the result. But with a divided party and the Conservative minority government fraying at the seams, either May or a new leader may choose to hold one sooner rather than later.
May had offered on Wednesday to resign if the deal passed, in a bid to win over eurosceptic rebels in her Conservative Party who support a more decisive break with the European Union than the divorce her deal offers.
Former Conservative Prime Minister John Major said: "When feelings are running high. a general election is pretty much the very last thing we need".
Speaking on Sky News, shadow foreign secretary Emily Thornberry said under a Labour government it was "likely" that the United Kingdom would leave the European Union.
Both were enormously controversial, long and hard political processes which nevertheless eventually succeeded and fundamentally changed the British political and social landscape.