Theresa May's Brexit plan is "not dead", a senior minister has insisted despite concessions made to Tory MPs to avoid a Commons defeat on trade.
Ministers have warned pro-European Tory MPs not to "refight the referendum" as the government faces a further test of its Brexit authority in the Commons.
The government scraped home by three votes on two occasions after agreeing to Brexiteers' demands to change the wording of the Customs Bill.
MPs are expected to vote later on calls by pro-European Tories for the UK to stay in a customs union if there is no trade agreement by 21 January 2019.
Liam Fox said it did not change policy as the amendments had been "cut and pasted" from the PM's Chequers plan.
The motion is being backed by Labour, making the risk of defeat more likely.
He also warned pro-European Tories against "refighting the referendum".
Trade Secretary Liam Fox said it would send the wrong message and urged MPs to rally round the PM's approach.
The international trade secretary told the BBC that feelings were running high but calls from some Tories to stay in a customs union, which will be voted on later, would send completely the wrong message to the EU.
The UK is due to leave the EU on 29 March 2019 but has yet to agree how its final relationship with the bloc will work.
The UK is due to leave the EU on 29 March 2019 but has yet to agree how its final relationship with the bloc will work.
The government, which does not have a Commons majority, has been under pressure from MPs on both sides of the Brexit debate.
The government, which does not have a Commons majority, has been under pressure from MPs on both sides of the Brexit debate.
The government twice survived by just three votes on Monday after a backlash from pro-EU Tories who accused the prime minister of "caving in" to the party's Eurosceptic MPs.
The government twice survived by just three votes on Monday after a backlash from pro-European Tories who accused the prime minister of "caving in" to the party's Eurosceptic MPs.
Fresh test ahead in Commons
Ministers accepted a series of demands from Brexiteers who are unhappy at the PM's Chequers blueprint for future relations with the EU, believing it keeps the UK too closely tied to the bloc.
Ministers accepted a series of demands from Brexiteers who are unhappy at the PM's Chequers blueprint for future relations with the EU, believing it keeps the UK too closely tied to the bloc.
But this angered MPs from the party's pro-EU wing who refused to back the new amendments, saying they would undermine the UK's recently-announced negotiating position.
MPs are continuing to debate Brexit on Tuesday as they consider the government's Trade Bill, which is entering its final parliamentary stage in the Commons.
By 305 votes to 302 - with 14 Tories rebelling - MPs backed an amendment that prevents the UK from collecting taxes on behalf of the EU, unless the rest of the EU does the same for the UK.
The legislation would create new structures for dealing with trade disputes after Brexit and carry over up to 40 existing EU trade agreements with other countries that the UK wishes to stick with until new deals are struck.
Applying EU tariffs to products destined for the EU is part of Mrs May's plan to avoid friction at UK borders after Brexit.
Call for 'safety net'
Another amendment, to ensure the UK is out of the EU's VAT regime, was backed by 303 to 300, with a Tory rebellion of 11. Three Labour MPs voted with the government. Current and past Lib Dem leaders Sir Vince Cable and Tim Farron - who want to stop Brexit - did not vote.
Ministers have, so far, comfortably defeated attempts by Labour and the SNP to give the UK Parliament and the Scottish Parliament an effective veto over future trade deals and for agreements to be subject to public consultation before they are ratified.
MPs will carry on debating Brexit on Tuesday when the Trade Bill comes to the Commons.
Attention is focused on a vote, expected before 18:00 BST, in which a handful of Tories are calling for a safeguards in the event of there being no trade agreement with the EU in the run-up to the UK's exit.
It gives the government the power to build new trade relationships around the world after the UK leaves the EU, and MPs who support staying in the EU's customs union are seeking to change its wording.
They have coalesced around a motion tabled by Tory Stephen Hammond which supports the government's aim of negotiating a free trade area for goods but states if this has not happened by 21 January, ministers must change tack and start discussions on joining a customs union.
'Strong feelings' on both sides
Labour MP Chris Leslie said a "safety net" was needed in the event of there being no deal as being outside a customs union would destabilise trade with countries such as Japan, as well as the EU.
Tory MP and Remainer Heidi Allen said she wished the prime minister had "faced down the amendments."
No 10 said Theresa May had been clear that leaving the customs union was necessary to enable trade deals with countries in other continents.
She told BBC Radio 4's Today programme: "What was agreed at Chequers wasn't perfect to us, wasn't perfect to Leavers either, but I think the prime minister had worked exceptionally hard to find a decent first pitch to put to the EU and move forward from that.
Mr Fox told Radio 4's Today compromises were necessary but any form of customs union was not acceptable and the referendum had given ministers the mandate to negotiate a deal which "reorientated the UK from a EU orbit to a global orbit".
"We were all set [on the Remain side] to drop all our amendments and back it, then suddenly we had these rather extreme last minute manoeuvres, which seem to us to deviate the prime minister from her plan and we weren't prepared to do that."
"I do not understand why people thinks this lacks democratic legitimacy," he said of the UK's strategy. "It is very clear where it comes from."
But Mr Fox said the amendments "did not differ very much" from the government's agreed position. Asked if the Chequers plan was dead, he told BBC Radio 4's Today "I don't think so".
Monday's controversial amendments, he argued, were different as they "did not differ very much" from the government's agreed position.
"The wording in the white paper was that the UK and the EU should together agree a mechanism for the remittance of relevant tariff revenue," he said.
"As far as I could see the amendment looked like a bit of a cut and paste from the white paper." Asked if the Chequers plan was dead, he replied "I don't think so".
"As far as I could see the amendment looked like a bit of a cut and paste from the white paper."
Former education secretary Nicky Morgan, who is backing Tuesday's customs union amendment, said she regretted that some of her colleagues seemed intent on trying "to make life difficult" for Theresa May.
He said the government could "not please everybody" and there had to be compromises but Brexit had been backed by 17.4 million people in a referendum and legislation implementing that decision approved by MPs.
"We just want to give the government space and the mood of the country is just to get on with it," she told BBC News.
"I do not understand why people thinks this lacks democratic legitimacy. It is very clear where it comes from."
Little room for manoeuvre
BBC political editor Laura Kuenssberg
It looks a mess because it is a mess. It's getting harder and harder for the prime minister to get things through Parliament - and while calls for a second referendum are widely rejected, that sentiment could change if this kind of gridlock continues.
The PM has spent the last two years trying to compromise. She has a divided party and no majority. There are no easy choices.
But the divisions in the Tory party are daily reducing her room for manoeuvre. In a debate about principle, the problem for some is that compromise is a dirty word.
Read Laura's blog
Who rebelled?
The Conservative rebels on Monday were the long-time pro-EU MP Ken Clarke, Heidi Allen, Guto Bebb, Richard Benyon, Jonathan Djanogly, Dominic Grieve, Stephen Hammond, Philip Lee, Nicky Morgan, Robert Neill, Mark Pawsey, Antoinette Sandbach, Anna Soubry and Sarah Wollaston.
The three Labour MPs who rebelled against their party whip by voting with the government were Frank Field, Kate Hoey and Graham Stringer - all of whom are pro-Brexit.
Former Labour MP Kelvin Hopkins who now sits as an independent also supported the government on one of the amendments.
How has the EU reacted?
BBC Europe editor, Katya Adler said the one priority the EU has is making sure it gets a deal, rather than a "cliff edge" Brexit.
She told Today: "They are following all the ins and outs, and all the turbulence, in UK politics extremely carefully.
"[But] they are wondering if the prime minister - or anyone who could or might take over from her - would even have the political strength to get a deal agreed here in Brussels, then passed by parliament back home.
"All my EU sources say they want to engage constructively with the whitepaper and avoid giving the impression that it is dead on arrival. But importantly, as everyone knows, time for negotiation is running short. They want to complete the withdrawal agreement."
