Theresa May and her cabinet are looking for ways to bring her EU withdrawal agreement back to the Commons for a fourth attempt at winning MPs’ backing.
The PM said the UK would need “an alternative way forward” after her plan was defeated by 58 votes on Friday.
MPs from all parties will test support for other options during a second round of “indicative votes” on Monday.
But government sources have not ruled out a run-off between whichever proves most popular and the PM’s Brexit plan.
Labour leader Jeremy Corbyn has called on Mrs May to change her deal or resign immediately, while Northern Ireland’s DUP – which has propped up Mrs May’s minority government – also continues to oppose the deal.
The government has so far failed to win over 34 Conservative rebels, including both Remainers and Tory Brexiteers who say the deal still leaves the UK too closely aligned to Europe.
However, a No 10 source indicated the prime minister would continue to seek support in the Commons.
They insisted efforts were “going in the right direction”, given the margin of defeat was down from 149 a fortnight ago.
Leave voters registered their anger at the latest rejection, on the day the UK was originally scheduled to leave the EU.
Thousands gathered outside Parliament to protest against the delay, bringing traffic to a standstill.
And the Conservative former Attorney General Dominic Grieve, who has campaigned for a further referendum on the deal, lost a vote of no-confidence in his Beaconsfield constituency.
Mrs May has until 12 April to seek a longer extension to the negotiation process to avoid the UK leaving without a deal, which most MPs believe could harm business and create disruption at ports.
However, she said any further delay to Brexit was “almost certain” to involve staging elections to the European Parliament in May.
Downing Street later said this was not an “inevitability” but Justice Minister Rory Stewart told BBC Newsnight Friday’s vote had been “the last chance” to avoid that.
He said it would take a “miracle” – and the support of up to 150 Conservatives – on Monday for a majority of MPs to back a Brexit option that supported staying in the customs union.
This allows businesses to move goods around the bloc without checks or charges but continued membership would bar the UK from striking independent trade deals.
And BBC political correspondent Chris Mason said: “Leaving it was a Conservative manifesto commitment, and an about-turn on that could tear apart the party from the cabinet down.”
Downing Street said Mrs May would continue to talk to the Democratic Unionist Party about more reassurances over the backstop – the “insurance policy” designed to prevent physical infrastructure at the Irish border.
The DUP says that by temporarily subjecting Northern Ireland to different regulations to the rest of the UK, the backstop would risk a permanent split.
Its Westminster leader Nigel Dodds told Newsnight: “I would stay in the European Union and remain, rather than risk Northern Ireland’s position. That’s how strongly I feel.”
And Conservative Mid-Norfolk MP George Freeman, who backed Mrs May’s deal, told the programme a cross-party solution was needed.
“The prime minister has run out of the road. We need to be setting up a Brexit war cabinet,” he said.
After the result of the latest vote was announced, Mr Corbyn said: “The House has been clear, this deal now has to change.
“If the prime minister can’t accept that then she must go, not at an indeterminate date in the future but now. So that we can decide the future of this country through a general election.”
Source: BBC