Eurosceptics believe it keeps Britain too close to the European Union, and Davis, the former Brexit secretary, warned in the Financial Times on Monday it would deny the government the "freedom to run our own economy".
But, if she chooses to fight and then sees a large number of her own party rebel, it would undermine her leadership and cast fresh doubt on whether she can deliver the Brexit plan agreed by her cabinet this month at her Chequers country residence.
She also stated that Britain would take a severe position in the next round of talks with the EU.
The votes in parliament's lower House of Commons come as Britain's new Brexit minister Dominic Raab heads to Brussels this week for his first talks with the EU's chief negotiator Michel Barnier. She endured a torrid week which began with the resignations of the Brexit Secretary David Davis and the Foreign Minister Boris Johnson.
The government's acceptance of the amendments did little to ease the tensions in May's party, which is at war with itself over the Brexit plans.
Rees-Mogg said ERG members would mount a show of strength on Monday evening, by voting for hard-Brexit amendments on the trade bill. "But in a trade sense, they've really taken advantage of us and many of those countries are in North Atlantic Treaty Organisation and they weren't paying their bills".
"Be in no doubt: under the Government's proposal our fingers would still be caught in this mangle and the European Union would use it ruthlessly to punish us for leaving and handicap our future competitiveness", Mr Davis said.
She said that would remove Britain's ability to have an independent trade policy and her government "will never stand for that".
The document affirmed that London was to withdraw from the EU's Single Market and Customs Union, ending the free movement and the jurisdiction of the European Court of Justice in the United Kingdom.
"This could lead to a damaging and disorderly Brexit", she wrote.
If no compromise is struck, there is a risk the eurosceptics could vote against the bill in its entirety.
In an interview published by The Sun, Trump said May's plan for the Brexit was counter to what "Leave" supporters approved in a referendum in 2016.
Rebellion from pro-Europe MPs in her party saw May narrowly avoiding defeat on two of the amendments by just three votes, 303 to 300, with the Prime Minister dependent on pro-Brexit opposition lawmakers to get the changes through.
"I was sitting down as a female leader with the President of the United States talking about issues that are of great importance to the people of the U.S., the people of the UK, but also actually of great importance around the world". "Actually, no, we're going into negotiations with them", May told the BBC in an interview. "That would be damaging to our "no deal" preparations".
Labour MP John Mann said some of his colleagues could back May's proposals.
Aside from the latest Brexit developments, Martin, a veteran Socialist MEP, said the time is ripe for a second referendum; "The general election proves there is no appetite for a hard Brexit - and now is the flawless time to change our minds".