The cross-partyproposal was the first time the House of Commons has held a formal vote on whether to endorse another referendum.
In 2016 Britain voted by 52 percent to 48 percent to leave the EU.
Parliament's votes this week won't end Britain's Brexit crisis.
May's spokesman said earlier on Thursday that she would put her Brexit deal, struck after two-and-a-half years of negotiations with the European Union, to another vote "if it was felt that it were worthwhile". "It was not in the best interests of the people of Wrexham or the rest of the country".
May's authority has been left severely damaged by a series of defeats and rebellions by her own ministers, including four Cabinet colleagues who failed to back the government in Wednesday's "no deal" vote.
Spectacularly close - Benn's vote to wrestle control off the government fails by just two votes, 312-314.
After a day of high drama, MPs defied the government by voting 321 to 278 in favour of a motion that ruled out a potentially disorderly "no-deal" Brexit under any circumstances.
A short extension could see the March 29 Brexit deadline kicked back somewhere between one and three months.
Brexit has become deadlocked in the British parliament, reflecting the deep divisions that remain in Britain nearly three years after the 2016 referendum to leave the European Union.
Now, MPs hope their 412 to 202 vote on Thursday will help them agree a deal.
In voting against Prime Minister Theresa May's divorce deal this week, hardline supporters of Brexit risked helping those who want to undermine the whole process.
PM Theresa May leaves 10 Downing Street ahead of the vote on delaying Brexit
The source warned that Mrs May had to tell Brussels her plans the moment MPs asked for an extension to give the EU-27 time to prepare their response for next week. They would probably want to throw us out, I should think. May has ruled that out, but could come to see it as her only option.
Lawmakers voted against May's amended brexit deal by 391 to 242 as her last-minute talks with European Union chiefs on Monday to assuage her critics' concerns ultimately proved fruitless. If there's no deal by March 20 - the eve of the next European leaders' summit - the delay will be much longer, she said.
The European Commission repeated that a delay would indeed require a justification 1 but positive comments from Germany and Ireland suggested that EU members at last saw a prospect that a viable deal would be found, and were inclined to help.
Officials and national leaders in the bloc are exasperated at the events in London.
A long extension could mean more time to work out a deal.
Bowing to pressure, deputy prime minister David Lidington said that if May's deal is not approved by next week, the government will "facilitate" votes in late March or early April "to seek a majority on the way forward".
In another sting for the beleaguered May, U.S. President Donald Trump said he was "surprised at how badly" the Brexit negotiations have been handled. "Politicians of every hue must overcome their differences and make avoiding a no-deal Brexit the absolute priority", she said.
The vote was not legally binding, however, and therefore does not rule out the possibility of the United Kingdom exiting the bloc without an agreement on the terms of its departure.
"There should be an overwhelming parliamentary majority to prevent a no-deal exit on March 29, which should lend some support to the pound", Morgan Stanley strategists said.
The bloc may be open to a long delay, however, to allow Britain to radically change course.