UK education minister Nadhim Zahawi warned that any delay, which meant fighting the European Parliament elections in May, would be a "suicide note" for the Tory Party.
He told BBC Radio 4's Today that the talks with the opposition are "ongoing and we hope that they deliver an outcome that allows Labour to support the Withdrawal Agreement".
He did add however that if Labour could not sign up to a joint approach, then MPs should be forced to find a compromise through a preferential voting system in order to resolve the situation before the European elections in May.
"If we can't get Labour to agree, I think it's important that Parliament acts quickly now to decide what it is in favour of - whether it is through some voting mechanism, where we have a single transferable vote and we vote on all the different deals.
"We need to do that quickly because I think going into the EU elections for the Conservative Party, or indeed for the Labour Party, and telling our constituents why we haven't been able to deliver Brexit I think would be an existential threat.
"I would go further and say it would be the suicide note of the Conservative Party."
Mr Zahawi added that the UK faced a "seismic" political shift with the rise of the hard-left and hard-right if Brexit was not delivered.
"We need to find out what Parliament will accept and then make a judgment as to whether that delivers Brexit," he told Today.
"If we do not deliver Brexit we would be unleashing forces that I think could get this country, and indeed the rest of Europe, into a very bad place."
Parliament's opposition to a no-deal Brexit meant "we are at the mercy of the 27 other countries" in the EU.
"If we do pass the Withdrawal Agreement and ratify it then we can leave. All other options will be ceding control to the EU27."
Meanwhile, UK finance minister Philip Hammond said he was optimistic over the deal the government is trying to strike with Labour.
As the talks between the British government and the main opposition party - aimed at breaking the deadlock over Brexit - continue, Mr Hammond said he expected to reach "some form of agreement."
The finance minister said the government had no red lines in the talks.
Arriving for a meeting of European Union finance ministers in Bucharest, Mr Hammond also said he was optimistic about the outcome of next Wednesday's EU summit on Brexit, as most EU states agreed there was a need to delay Brexit, as requested by British Prime Minister Theresa May.
In her response, Labour's shadow home secretary Diane Abbott said: "We engaged in these talks in good faith. Keir Starmer has written to the Government to say he wants to continue the talks, so in that sense they are going on.
"There is concern that the Government doesn't want to alter the Political Declaration."
She told BBC Radio 4's Today programme: "There's no question that the mess we are in is Theresa May's mess, even Tory MPs accept that.
"The Labour Party has stepped up, we want to help. We are engaged in these talks in good faith but the government perhaps has to show a little more flexibility than it seems to have done so far."
Labour's position was that it wanted a customs union, single market alignment, protection of rights and "some kind of People's Vote".
Pressed on whether any agreement must have a second referendum attached to it, she said: "We are not saying anything definitively but we have a position."
While it "has to be part of the negotiations", Ms Abbott said "we have not gone into these talks being dogmatic".