A proposal to hold a confirmatory referendum on any deal got the most votes, but was defeated by 292-280.
Motion D, surrounding a Common Market, was defeated by 282 to 261 votes.
To be granted a further postponement the Government will have to set out what goal it would achieve and it will nearly certainly involve taking part in the European elections.
The EU has called an emergency summit for April 10 and warned that without a plan, Britain risks abruptly ending ties with its largest trading partner two days later, causing huge economic disruption.
Labour leader Jeremy Corbyn speaking in the House of Commons, London after MPs fail to back proposals on alternatives to Theresa May's European Union withdrawal deal. She is under vast pressure from both party wings to decide whether to put her own deal forward for a fourth vote, or switch to a plan B. But she has been warned that ministerial resignations will be "unavoidable" if she picks either no deal or a customs union.
Four options were presented to MPs in the House of Commons yesterday but all were rejected late last night.
The final option, which would have instructed government to revoke the legislation to leave the European Union a day before Britain is due to crash out, was rejected by 292 to 191.
Meanwhile, fellow Conservative MP, Nick Boles announced his resignation from the party whip after seeing his proposal for the United Kingdom to stay in the EU's single market defeated by a 21 majority, stating that he had "failed chiefly because my party refuses to compromise".
The analysts at MUFG said that while a no-deal Brexit on April 12 remained a risk for the pound, an emergency European Union summit on April 10 could - if British lawmakers cannot agree anything this week - see a long delay to Brexit.More news: Korea's Moon to meet Trump in US on nuke diplomacy：The Asahi Shimbun
"This house has continuously rejected leaving without a deal just as it has rejected not leaving at all", Mr Barclay said.
Speaking just minutes after the voting results came out, he said: "I have failed, chiefly, because my party refuses to compromise".
Oliver Letwin, the Conservative lawmaker behind the process of votes, has previously said that if parliament succeeded in finding a majority for a way forward, he hoped the government would accept that outcome but if it did not, then lawmakers would bring forward legislation seeking to force it to do so.
The process was launched after MPs voted to take control of the Brexit agenda from the government after repeatedly defeating May's deal.
Marshall Gittler, a strategist at ACLS Global, said he considered a no-deal Brexit "as a higher possibility, even though it's officially been ruled out, simply because I don't see any of the other endings as particularly possible".
However, bringing back the vote for the fourth time could prove hard, given that the House of Commons speaker has suggested he would block any move for the deal to be voted on again.
The government will make clear its plans for another so-called meaningful vote after Monday evening's Commons vote on different Brexit options.
On Monday, it was revealed the Conservatives chief whip, in charge of party discipline, had heavily criticised the Cabinet's behaviour.