British Prime Minister Theresa May on Monday refused to back down on inviting US President Donald Trump for a glitzy state visit, despite a petition gaining more than 1.5 million signatures and protesters rallying across the country.

May is standing firm on the invitation, which would see Trump honoured by parliament and Queen Elizabeth II, despite the premier's office saying she does not agree with his controversial ban on refugees and citizens of seven mainly Muslim countries.

"The United States is a close ally of the UK, we work together across many areas of mutual interest and we have that special relationship between us," May told a press conference in Dublin alongside her Irish counterpart Enda Kenny.

"I have issued that invitation for a state visit to President Trump to the UK and that invitation stands."

The British premier added: "In relation to the policies that have been announced by the US, the UK takes a different approach."

May announced the state visit during a meeting with Trump at the White House on Friday, which was intended to boost trade ties after Britain leaves the European Union.

But her closeness to the erratic billionaire drew fire at home, particularly when she initially failed to condemn the travel ban, saying it was a US affair.

- Tens of thousands protest -

Protesters rallied in cities across the UK on Monday, with a police source telling AFP the crowd in London stood at tens of thousands.

"Don't hold hands with Nazis" and "No to racism, no to Trump" read placards in London, while the crowd chanted: "Shame on you Theresa May."

Protester Annette Connors, 41, said the government's position on Trump suggested Britain was more concerned with scoring a trade deal than principles.

"I'm very much against Trump being invited for a state visit, I'm also very disappointed at the government response to what he does compared to European leaders' responses," she told AFP.

In Manchester, northwest England, crowds chanted "no state visit" and "refugees are welcome here".

Janet Harbour, 54, said she was protesting for the very first time.

"He is a disgrace to America," she told AFP. "America can do better and needs to do better."

The national mood was captured on several of Britain's newspaper front pages for Tuesday, many of which carried protest photos contrasting with May's position.

By Monday evening, more than 1.5 million people had signed an online petition calling for the state visit to be cancelled because Trump's "well-documented misogyny and vulgarity disqualifies him" from meeting the queen.

Set up several months ago, the number of signatures has surged since Trump signed his contentious executive order on Friday.

- 'Divisive and wrong' -

Opposition Labour leader Jeremy Corbyn, the smaller Liberal Democrats and the Scottish National Party have all called for the state visit to be cancelled, as have some from May's own party.

Conservative Muslim lawmaker Sayeeda Warsi told BBC radio that Britain should question whether it should roll out the red carpet for "a man who has no respect for women, disdain for minorities... and whose policies are rooted in divisive rhetoric."

Trump's order suspended the arrival of all refugees for at least 120 days, Syrian refugees indefinitely, and barred citizens from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen for 90 days.

British foreign minister Boris Johnson told parliament it was "divisive and wrong to stigmatise because of nationality", confirming that, after talks with the US administration, most Britons would be exempt.

However, Johnson said there was "no reason" why Trump shouldn't receive a state visit, calling the US-UK relationship the "single most important geopolitical fact of the last 100 years". afp