The Prime Minister has defended her invitation to honour Mr Trump with a state visit despite a growing outcry, with more than 1.6 million people signing a petition calling for it to be scrapped.
However, she is facing calls from MPs to say what she was told by American officials about the temporary ban on nationals from seven mainly Muslim countries.
The order was issued hours after her meeting with the president in the White House on Friday.
Downing Street refused to be drawn on a report by Channel 4 News that she had been told refugees would be barred from travelling to the US, although officials were said not to have revealed much detail.
“You will have heard the Prime Minister and the president’s comments following their discussions and we are not going to go into details of a private meeting,” a No 10 spokesman said.
On Monday, Foreign Secretary Boris Johnson told MPs he was not prepared to comment on “confidential conversations” between the two leaders.
Liberal Democrat leader Tim Farron said it was “disgraceful” that Mrs May had appeared to know about the ban in advance but did nothing to prevent it.
He said: “I can only assume the Prime Minister is so desperate for a Brexit deal that she looked the other way and didn’t want to rock the boat.
“This is utterly shameful. Parliament needs to know what she knew and when.”
Despite growing protests at the planned state visit, Mrs May - in Dublin on Monday for talks with Irish prime minister Enda Kenny - was adamant it would go ahead.
However, a former head of the Foreign Office warned the Prime Minister’s decision had put the Queen in a “very difficult position”.
Lord Ricketts, who was permanent secretary at the Foreign Office from 2006 until 2010 before becoming David Cameron’s national security adviser, said the offer so early in Mr Trump’s presidency was “premature”.
In a letter to The Times, he said it was unprecedented for a US president to be given a state visit in their first year in the White House and questioned whether Mr Trump was “specially deserving of this exceptional honour”.
“It would have been far wiser to wait to see what sort of president he would turn out to be before advising the Queen to invite him. Now the Queen is put in a very difficult position.”
In the Commons, Mr Johnson told MPs the Government had been given assurances the ban would not affect British passport holders.
Foreign Office sources suggested the UK had secured a “special carve-out” from Mr Trump’s policy after a round of frantic diplomatic activity, with Mr Johnson and Home Secretary Amber Rudd contacting their counterparts in the US.
The US embassy had earlier suggested that UK citizens with dual nationality including one of the seven countries covered by the temporary travel ban - Iraq, Iran, Libya, Somalia, Sudan, Syria and Yemen - should not seek to obtain a visa.
The guidance was subsequently removed from the embassy website.
Former foreign secretary Lord Hague said he would have preferred the first ten days of Mr Trump’s presidency to have been “a bad dream from which we can now wake up”.
However, writing in the Daily Telegraph, he said Mrs May had “played a blinder” in offering the state visit to the president in order to secure agreement on trade and Nato.
Retracting the invitation would only damage Britain’s credibility, Lord Hague warned.
As for causing any embarrassment to the Queen, he said: “A Queen who has been asked over the decades to host tyrants such as Presidents Mobuto of Zaire and Ceausescu of Romania is going to take a brash billionaire from New York effortlessly in her stride.”