Theresa May is appealing for MPs to be "patriotic" and vote for her Brexit deal this week.
It is reported more of her backbenchers are set to support her, and even the DUP could be on the brink of backing her.
Mrs May called on politicians to look beyond the categories of Leave and Remain and "resolve this question now". If she moves ahead with this third "meaningful vote", it is expected it could take place on Tuesday.
Sky's political reporter Rob Powell says there has been a slow trickle of Brexiteers coming round to her deal.
Tory former Cabinet minister Esther McVey, who has said she will now back Theresa May's Brexit deal, suggested the Prime Minister should consider setting out the timetable for her departure in order to win over other rebels.
"She will have to think about that herself," Ms McVey told Sky News' Sophy Ridge On Sunday. "She has got to look at the fact - when collective responsibility collapses like that, who is the voice of government?
"The Government should only have one voice so the country knows what we stand for, so the world knows what we stand for."
She said Mrs May's Cabinet was "making her position very dubious".
Theresa May has warned MPs that if they fail to back her Brexit deal at the third time of asking then Brussels might insist on a lengthy delay, potentially scuppering chances of leaving the European Union altogether.
Mrs May said it would be a "potent symbol of Parliament's collective political failure" if a delay to Brexit meant the UK was forced to take part in May's European elections almost three years after voting to leave.
She warned 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".
Mrs May acknowledged that even if her deal is passed before the summit of EU leaders the Government would need a "short technical extension" beyond the scheduled March 29 Brexit date.
"That is not an ideal outcome - we could and should have been leaving the EU on March 29," she said.
"But it is something the British people would accept if it led swiftly to delivering Brexit. The alternative if Parliament cannot agree the deal by that time is much worse."
Writing in the Sunday Telegraph, she said her fellow leaders would demand a "clear purpose" for a longer extension that was not merely a technical delay to allow legislation to pass.
In a message aimed at MPs seeking rival versions of Brexit, she said: "If the proposal were to go back to square one and negotiate a new deal, that would mean a much longer extension - almost certainly requiring the United Kingdom to participate in the European Parliament elections in May.
"The idea of the British people going to the polls to elect MEPs three years after voting to leave the EU hardly bears thinking about.
"There could be no more potent symbol of Parliament's collective political failure.
"All this makes the choice now facing MPs clearer than it has ever been. If Parliament can find a way to back the Brexit deal before European Council, the UK will leave the EU this spring, without having to take part in the European elections, and we can get on with building our future relationship with the EU.
"If it cannot, we will not leave the EU for many months, if ever."
In an appeal to MPs to back her deal, she said: "I know that I will have to do more to convince others, as well as the DUP, if I am to succeed in finally securing a majority for the deal.
"That remains my goal. It is the only way through the current impasse."
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.
Backbench rebel Daniel Kawczynski publicly indicated he will now back Mrs May's plan.
But so far the number of Tories publicly switching positions has amounted to a trickle rather than the flood she needs to overturn the 149-vote defeat for the deal she suffered on Tuesday.
MP Mr Kawczynski acknowledged that the Prime Minister's deal was now the "only game in town".
His comments came after former cabinet minister Esther McVey, who resigned over the Brexit deal, suggested she and other MPs could now back it, even though it was "rubbish" and North Wiltshire MP James Gray appealed to fellow members of the Brexiteer European Research Group (ERG) to get the "obnoxious" deal over the line.
But other ERG figures played down the prospect of large numbers getting behind the deal when it is brought back for a vote.
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."
Meanwhile, talks continued with the DUP, but the party stressed that the presence of Chancellor Philip Hammond at a meeting on Friday did not mean that money was being demanded.
"We are in discussions with the Government to ensure Northern Ireland is not separated out from the rest of the United Kingdom as we leave the European Union," a spokesman said.
"Contrary to some reports we are not discussing cash...There are still issues to be addressed in our discussions."
Jeremy Corbyn has offered talks with opposition leaders and backbench MPs in an effort to find a Brexit compromise which could replace Mrs May's plan.
Labour is expected to throw its weight behind an amendment tabled by backbenchers Peter Kyle and Phil Wilson which would make support for the Brexit deal conditional on a referendum.
Meanwhile, International Trade Secretary Liam Fox said Theresa May might not hold another vote on her Brexit deal this week if she is on course to another defeat.
Mr Fox told Sky News: "That would be determined by whether we can succeed in getting that vote through the House of Commons."
He said it would be "difficult to justify having a vote if you knew we were going to lose it".
He added: "I would say to my colleagues: all actions have consequences, and if you really want to deliver the Brexit we all promised... if we actually want to do that then we need to back the Prime Minister's deal because there is no other deal on offer."