Clashes centred on staying in a customs union with Brussels continue to dog the Government, with flagship Brexit legislation vulnerable to further attacks in the House of Lords.
Pro-European Tory MPs are set to voice their concerns over the planned withdrawal on Thursday.
But Downing Street sources insisted there would be no backsliding over quitting the customs union with Brussels after reports that the Prime Minister was ready to take a softer line.
Though the looming Commons vote on a pro-customs union motion would be a symbolic, non-binding one, it has the potential to deepen Tory wounds on Brexit.
Mrs May is set to face calls from leading Brexiteers Boris Johnson, David Davis and Liam Fox to abandon her preferred form of a customs deal with the EU, according to The Times.
A showdown is expected to come at a meeting of the Cabinet Brexit Committee scheduled for Wednesday when the trio will tell the PM that a so-called customs partnership, where the UK would collect EU import tariffs on behalf of Brussels, would be unworkable, the report said.
Former Cabinet minister John Whittingdale told BBC Radio 4's The Westminster Hour that defeats like the one the Government suffered last week in the Lords over the customs union played into the EU's hands.
"Part of the problem is that there are some in the European Union and in the Irish Republic who want to make out that we have to stay in the customs union and will not examine other alternatives.
"And votes like ones in the House of Lords are going to add to the view over there that somehow the British Government might U-turn or go back.
"We need to make it absolutely clear that leaving the customs union is an absolutely fundamental part of Brexit. We then have to find a solution to the Northern Ireland border question."
Labour MP Chuka Umunna and backbench pro-Remain Tory Anna Soubry have tabled amendments to the Trade and Customs Bill, due to be debated on Thursday, that will make staying in the customs union a legal objective of the Government.
Mr Umunna said on Twitter: "The Brextremists threaten May with a leadership election if she concedes on the customs union but a new Tory leader would face the same Parliamentary arithmetic."
Cabinet minister Sajid Javid tweeted that the referendum gave "clear instructions" to leave the customs union and accused some of seeing it as a "kind of post-Brexit comfort blanket".
The Housing and Communities Secretary's comments were rebuked by the head of the CBI, Paul Drechsler, who tweeted in reply: "An MP of your talent should rise above ideology and lead based of fact, analysis and evidence - all of which favours a customs union. Always happy to discuss."
The Lords could deliver new defeats on the Government regarding incorporating the EU charter of fundamental rights into UK law, and pressing to ensure MPs get to decide what happens next if the Commons rejects a final Brexit deal with Brussels.