Theresa May could duck a third Commons showdown on her Withdrawal Agreement, resulting in a long delay to Brexit unless Tory rebels fall in line.
Cabinet ministers said another vote on the deal would not be held unless the Prime Minister was confident of overturning last week's overwhelming defeat.
Unless the 149-vote humiliation can be reversed, Mrs May would be forced to seek a lengthy extension to the Article 50 process, with the UK required to elect MEPs in May, Chancellor Philip Hammond said.
The Prime Minister herself has warned of what Westminster insiders have dubbed a "Hotel California" Brexit where the UK can never leave.
She said that if MPs did not back her deal before Thursday's European Council summit "we will not leave the EU for many months, if ever".
It would be a "potent symbol of Parliament's collective political failure" if a delay to Brexit meant the UK was forced to elect MEPs in May almost three years after voting to leave, Mrs May said in a Sunday Telegraph article.
As part of the effort to put pressure on Tory hardliners and the DUP to shift position and support the deal, Mr Hammond and International Trade Secretary Liam Fox made clear that the Prime Minister might not even risk a third defeat on her deal after the 230-vote loss in January and last week's 149-vote reverse.
Mr Hammond told BBC One's Andrew Marr Show: "We will only bring the deal back if we are confident that enough of our colleagues and the DUP are prepared to support it so that we can get it through Parliament.
"We are not just going to keep presenting it if we haven't moved the dial."
He said a "significant number" of MPs had changed their minds because the alternatives to Mrs May's deal were "so unpalatable" to them, but he acknowledged it remained a "work in progress" to secure enough votes.
But in a warning of what was at stake Mr Hammond added: "Clearly if we don't get this deal through we are almost certainly going to have to fight a European parliamentary election, almost certainly going to have a longer extension, almost certainly not going to be able to gain access to the fiscal headroom I talked about in the Spring Statement."
The Chancellor insisted it was "not economic blackmail, it's common sense" to say that he would not be able to release the almost £27 billion of "fiscal headroom" unless the deal was done, as he needed to keep it in reserve to cope with a no-deal departure.
Dr Fox told Sky News' Sophy Ridge On Sunday: "I would say to my colleagues: all actions have consequences, and if you really want to deliver the Brexit we all promised... then we need to back the Prime Minister's deal because there is no other deal on offer."
He added: "If we had an extension with no agreement and this was just kicking into the long grass with the chance that Brexit might not happen at all, that would be very, very hard for most people to swallow."
The possibility of Brexit being delayed or overturned in a second referendum is swinging some Eurosceptics reluctantly behind the deal ahead of a third vote on the package.
Former Cabinet minister Esther McVey told Sky News that, although it remained a "bad deal", "the choice before us is this deal or no Brexit whatsoever".
Shrewsbury and Atcham MP Daniel Kawczynski has also acknowledged that the Prime Minister's deal was now the "only game in town".
Former Brexit secretary David Davis said the PM's deal was "capable of rescue" if the negotiating strategy was changed and experienced trade negotiators were brought in to work with the EU.
Writing in The Sunday Times, he added: "Judging by last week the alternative would be a cascade of chaos, eventually ending in a remainer attempt, first to delay Brexit by a long time and then reverse it."
Former Vote Leave chief executive Matthew Elliott, writing in the same paper, called Mrs May's deal "far from perfect", but added: "I would rather opt for the risk of a customs union later — a risk that has diminished in recent weeks — than the very real risk of a permanent customs union now. The choice isn't enviable, but the safer option is clear."
But so far the number of Tories publicly switching positions has amounted to a trickle rather than the flood the Prime Minister needs as she seeks 75 more votes for her deal.
Constituents fed up with infighting in both Parties.Parliament voted this week to take no deal from table (mistake in my view)) No other deal can get through Commons so Mrs May's Deal is only game in town. Do we continue to obstruct & risk no Brexit? I doubt I can take that risk
— Daniel Kawczynski (@DKShrewsbury) March 16, 2019
Meanwhile, talks continued with the DUP, but the party stressed that the presence of Mr Hammond at a meeting on Friday did not mean that money was being demanded.