Backbenchers in parliament's lower House of Commons seized the initiative by holding a round of votes last week on eight alternative Brexit options, but failed to agree on any of them.
A vote on plans for a second referendum went down by 292 to 280.
The result was close for proposals to negotiate a permanent customs union with the EU. This motion saw 276 votes against it and 273 votes for it.
As MPs finally just about managed to decide something - not to have any more votes created to find out what kind of Brexit they could actually support - and the Governor of the Bank of England warned of the "alarmingly high" risk of an "accidental disorderly Brexit", Theresa May and Jeremy Corbyn held a meeting.
The latest twist in the Brexit saga came after one Kent MP said she understood the frustration felt by a former Conservative MP Nick Boles.
"We are now in a really risky situation with a serious and growing risk of no-deal in 10 days' time", said opposition Labour lawmaker Yvette Cooper, who has proposed the legislation alongside eleven others from several political parties, including members of May's Conservatives.
The draft legislation by the former Labour minister would force the prime minister to ask the European Union for an extension to the Article 50 process beyond 12 April and would give Parliament the power to decide the length of this delay.
May had told her ministers to boycott the indicative votes.
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After the vote, British Steven Barkley, the UK's secretary of state, said the United Kingdom could avoid a prolonged postponing of the Brexit process if parliament accepts the deal for Brexit this week, Reuters reported.
After failing repeatedly to win Parliament's backing for her Brexit blueprint, May said the country needed "national unity to deliver the national interest". "More today than ever".
After a tumultuous week in which May's divorce strategy was rejected by lawmakers for a third time, despite her offer to quit if it passed, the future direction of Brexit remains mired in confusion.
The European Union has already called for a summit on April 10 to discuss any last minute proposals for an extension of the deadline if a deal is agreed by the British MPs. The government as expressed by the Prime Minister is against no-deal.
Theresa May can not ignore backing for a "softer" Brexit.
The April 12 deadline, imposed by the European Union, gives Britain's politicians less than two weeks to bridge the hostile divide that separates those in her government who want to sever links with the European Union and those who want to keep the ties that have bound Britain to the bloc for nearly 50 years. As things stand, Britain will now depart at 10 p.m. on April 12 - unless May comes up with another viable option.
MPs met in the House of Commons for a whole day of debate and voting on Brexit - but weren't able to agree a way forward.
Conservative Party MPs are furious and threatening to block a snap election in which many could lose their seats.