Downing Street said its position had not changed on the US president's trip as British Prime Minister Theresa May came under pressure to downgrade or axe the visit.
"An invitation has been extended and accepted," a Number 10 spokesman said, stressing the position had not changed.
The Foreign Office said Britons with dual citizenship would be exempt from the controversial travel ban, but UK residents have been caught up in the chaos.
The US president accepted an invitation to visit Britain later this year, where he is due to be hosted by the Queen and would be treated to all the pomp and ceremony accorded to a state visit.
But nearly one million people have signed a petition stating Trump should not be given a state visit and it will now be considered for debate by MPs.
Labour, the Liberal Democrats and the SNP have called for the lavish state visit to be cancelled and Tory MPs have warned against allowing it to go ahead.
Conservative former foreign office minister Alistair Burt said the "optics of a visit are currently very bad" and suggested American officials should find a way for it not to go ahead.
Labour MP Dan Jarvis said it was "very likely" there would be an urgent question in the House of Commons later on Monday to discuss the travel ban.
Demonstrators are planning to protest against the policy outside Downing Street and across the UK on Monday night.
The US president's team told Foreign Secretary Boris Johnson that Britons who have shared nationality with one of the seven mainly-Muslim countries covered by the restrictions would not be stopped from entering America.
But UK dual citizens travelling to the United States directly from one of the banned countries will face extra checks.
Johnson spent Sunday speaking to the president's senior adviser Jared Kushner and chief strategist Stephen Bannon about the implications of immigration curbs.
