Theresa May headed to Wales and Northern Ireland, hours after Trump said it seemed like a "great deal" for the EU that could block Britain from striking its own trade agreement with the United States.

May has two weeks to convince the public, and, crucially, a divided parliament, before a vote in the House of Commons on December 11 that risks ending in a humiliating defeat and sinking the deal.

Trump warned that the terms of the deal might block a future trade deal between London and Washington and suggested May had made a mistake.

"Sounds like a great deal for the EU," he said at the White House, adding: "we have to take a look at seriously whether or not the UK is allowed to trade".

"As the deal stands, they may not be able to trade with the US and I don't think they want that at all. That would be a very big negative for the deal," said the president, who is close to leading Brexiteers in the UK.

"I don't think that the prime minister meant that and hopefully she'll be able to do something about that."

- Downing Street strikes back -

May's Downing Street office hit back at Trump's comments, saying that Britain would be free to strike its own trade agreements outside the bloc.

"We will have an independent trade policy so that the UK can sign trade deals with countries around the world -- including with the United States," a spokesman said.

"We have already been laying the groundwork for an ambitious agreement with the US."

The pound was down almost half a percent against the dollar and lower also against the euro in the wake of Trump's comments. London's FTSE 100 shares index was up 0.1 percent.

May on Sunday closed 17 months of complex talks with Brussels by sealing Brexit arrangements with the 27 other EU heads of state and government.

But this tortuous chapter on ending Britain's 45-year involvement in the European project was just the beginning.

May runs a minority Conservative government and opposition parties, not to mention many of her own MPs, are against the deal.

Some Brexiteers think it keeps Britain shackled to Brussels while pro-EU lawmakers think the terms are worse than staying in the bloc and want a second referendum.

Opposition Labour party leader Jeremy Corbyn called the deal "an act of national self-harm".

- TV face-off -

May, who ducked televised debates during the 2017 snap general election, challenged Corbyn to a TV contest, nominally planned for December 9.

"I am ready to debate it with Jeremy Corbyn because I have got a plan. He hasn't got a plan," May was quoted as saying by The Sun newspaper.

A Labour spokesman said: "Jeremy would relish a head to head debate with Theresa May about her botched Brexit deal and the future of our country."

When May defended the deal in parliament on Monday, more than an hour passed before a Conservative voiced support for the agreement.

In a sign of the difficulties facing May, former defence secretary Michael Fallon, once an ultra-loyalist, said Tuesday that the deal gave Britain "the worst of all worlds" and vowed to vote against it on December 11.

He said "the deal is doomed" unless MPs can "be persuaded somehow" that it guarantees the ability to reduce tariffs and strike trade deals with countries outside the EU.

"If it's possible to get a better deal" by postponing Brexit for three months, "in the long term that would be in the best interest of the country", said Fallon.