Ministers are trying to rally support for the PM's Brexit deal across the UK but a bid to find a compromise has been dismissed by the DUP and Brexiteers.
With Theresa May widely expected to lose Tuesday's Commons vote, No 10 has dismissed calls for it to be delayed.
Health Secretary Matt Hancock said with two days of debate remaining, there was still time to win over MPs.
But a Tory backbench amendment aimed at easing concerns about the controversial "backstop" has met with criticism.
Downing Street has said the vote is still due to take place on Tuesday, despite dozens of Tories threatening to reject the deal, along with the DUP, whose support keeps Mrs May's government in power.
But a senior minister has told the BBC "the only political common sense is to delay" it.
The minister, who preferred not to be named, said: "We need to find a solution and we can't find one by Tuesday."
Matt Hancock, Chancellor Philip Hammond, Cabinet Office Minister David Lidington and Scottish Secretary David Mundell are among those trying to sell it to the public in visits across the UK.
Amid calls from some Tory and Labour MPs for the UK to pursue an even closer Norway-style relationship with the EU, Mr Lidington said it was wrong to assume there was a "magic alternative" waiting in the wings which retained the existing benefits of membership without the obligations.
"If it is not this deal which the rest of the EU says they are not willing to renegotiate, then either you crash out of the EU without any deal, without any transitional period or you revisit the referendum result of 2016 and you stay in the European Union," he said.
The withdrawal deal negotiated between the UK and EU has been endorsed by EU leaders but must also be backed by Parliament if it is to come into force.
Many MPs have expressed concerns about the backstop, aimed at preventing a "hard border" between Northern Ireland and the Republic of Ireland, which will remain in the EU, if no trade deal is ready before the end of the post-Brexit transition period.
It would mean Northern Ireland staying aligned to some EU rules, which many MPs say is unacceptable. The UK would also not be able to leave the backstop without EU agreement.
Jeremy Corbyn told Euronews that in a Labour Brexit deal "there certainly wouldn't be a backstop from which you can't escape".
"We will have to come to an agreement on a customs union, a specific customs union with the European Union that does give us the opportunity to have a say in it all, but also guarantees that level of trade," the Labour leader said.
Downing Street has dismissed reports the vote could be delayed. And Mr Hancock told the BBC that "the best thing for the country" was for MPs to back Mrs May's deal.
"My view is we should continue the debate," he said. "We've had three days, there's two days more. I think we should make the argument, make the case and persuade people - that's what you have Parliamentary debate for."
Speaking to BBC Radio 5 live, Education Secretary Damian Hinds acknowledged the government had "a big job on" to win Tuesday's vote, but appealed to MPs to back the deal "in the national interest". And Chief Secretary to the Treasury Liz Truss said while the deal on offer was "not perfect" - it delivered on what people voted for.
The prime minister has suggested that MPs could be "given a role" in deciding whether to activate the backstop, and on Thursday night, a Tory backbench amendment was laid down intended to do that.
The amendment - which is understood to have government support - would also give the devolved administrations, particularly the Northern Ireland Assembly, although it is currently suspended, more say in the process.
It would also press the UK and EU to agree a future trade deal within a year of the implementation period ending.
Former Northern Ireland minister Hugo Swire tabled the amendment along with Bob Neill and Richard Graham.
Mr Swire told the BBC that many Tory MPs would like to see the backstop "disappear altogether or be time limited" but the European Commission had said it would not reopen negotiations on the withdrawal agreement, so the amendment was "about the nearest we feel we can probe".
He said it was aimed at "people like me, who would like to be able to support this deal but find they are unable to".
But Conservative Brexiteer Steve Baker dismissed the amendment: "Giving Parliament the choice between the devil and the deep blue sea is desperate and will persuade very few."
And fellow backbencher Peter Bone told the BBC the amendment was "absolutely meaningless".
DUP leader Arlene Foster tweeted: "Domestic legislative tinkering won't cut it. The legally binding international withdrawal treaty would remain fundamentally flawed, as evidenced by the attorney general's legal advice."
Meanwhile, former cabinet minister Justine Greening has told the BBC's Political Thinking podcast that a Conservative Party "that seems consumed by Brexit" would lose the support of "Middle England".
"One of the problems for the Conservative Party is it's now 31 years since we last won a landslide and we need to realise as a party if things don't change, that'll be our last landslide," she said.