Prime Minister Theresa May is writing to the EU to formally ask for Brexit to be postponed. One ministerial source told the BBC the longer delay could be up to two years, amid reports of a cabinet row, but No 10 said no decision had been made.
EU Brexit negotiator Michel Barnier said the EU would not grant a delay without a concrete plan from the UK about what they would do with it.
Under current law the UK will leave the EU with or without a deal in 10 days.
MPs rejected the withdrawal deal Mrs May has negotiated with the EU for a second time last week by 149 votes. They also voted in favor of ruling out leaving the EU without a deal, and in favor or extending the Brexit process.
The prime minister had hoped to have another try at getting MPs to back the deal this week - but Speaker John Bercow effectively torpedoed that with his surprise intervention on Monday.
She still hopes to ultimately get it in front of MPs for a third go, but says even if that happens and they vote in favor of it, the UK will need a short extension to get the necessary legislation through Parliament.
A cabinet source told the BBC she therefore plans to ask the EU to agree to postpone the UK's departure until 30 June, but with an option of a longer delay as well.
Mrs May has warned Brexiteer Tories that longer extension will be needed if her deal does not get through Parliament.
However cabinet discussions were strong with Commons leader Andrea Leadsom is said to have criticized colleagues, saying they now amounted to a Remain cabinet, not a Brexit cabinet.
But Chancellor Philip Hammond said ministers all wanted the shortest possible delay but cabinet members have different approaches to how we should do this.
Any delay will have to be agreed by all 27 EU member states and Mrs May is heading to Brussels on Thursday to discuss the options with fellow leaders.
Brexit Secretary Stephen Barclay suggested a third vote on the Brexit deal could take place next week - after Mrs May has sought a delay.
On Mr Bercow's ruling, he said it was important to respect the referee and abide by his decisions - but, he added, the Speaker himself had said in the past that if Parliament was guided only by precedent then nothing ever would change.
Mr Barclay suggested that MPs would find a way to get another vote, if the government manages to persuade enough of them, including the 10 Democratic Unionists, to change their mind and back the deal.
On Monday, the Speaker said he would not allow a third meaningful vote in the coming days on substantially the same motion as MPs rejected last week.
Meanwhile Labour leader Jeremy Corbyn said the Speaker's intervention showed Mr Bercow was ensuring Parliament is taken seriously.
He said he had spoken to Conservative and Labour MPs about a so-called Norway-Plus style of future relationship with the EU - a closer one than Mrs May's deal would bring about - calling it an interesting idea which had not got his complete support.
When asked about another referendum, he said: The issue has to be put to the people after Parliament has made some kind of decision.
He said a public vote had to be on a range of options, saying: It cannot just be on Theresa May's deal or Remain - there has to be some choice for the people.
But Liz Saville Roberts, Plaid Cymru's Westminster leader, suggested the Labour leader had not really engaged during the meeting with the case for giving the public the final say on Brexit.
We were there to talk about a People's Vote and the only thing I felt he was comfortable talking about was Labor's version of Brexit, she said.