Trump said on Monday the agreement by which the United Kingdom will leave the European Union may make trade between Washington and London more hard - handing ammunition to May's critics at home who want to block the Brexit deal.
On Monday, she came under fire from MPs over her Brexit deal in the House of Commons.
"I am willing to stand up and explain why I think it is the best possible deal available for the United Kingdom".
"Because, you know, right now if you look at the deal, they may not be able to trade with us and that wouldn't be a good thing", added the president.
The prime minister faces an uphill struggle to get the negotiated withdrawal agreement and political declaration through the Commons in the so-called "meaningful vote" next month.
"No guarantee of smooth trade in the future and no ability to reduce the tariffs that we need to conclude trade deals with the rest of the world", Fallon told BBC radio.
The groundwork for an "ambitious" agreement with the U.S. has been laid in five meetings of joint working groups, the spokesman said.
May returned to the lower chamber of lawmaking a day after European Union member states gave unanimous backing to a deal on the UK's withdrawal from the bloc but now must convince a majority of Members of Parliament to vote in its favor when it is put to a meaningful vote in the chamber.
"For the good of the nation, the House has very little choice but to reject this deal", Corbyn said.
May's deal for Brexit was accepted by the leaders of the European Union on 25th November but still has to get the approval of parliament.
He said the Brexit deal, endorsed Sunday by European Union leaders in the special Brexit summit in Brussels, betrayed the Brexit people voted for in the 2016 national referendum, adding that the House of Commons would have to reject her deal.
There will be 5 days of debate before MPs will decide whether to back May's deal.
However, many British MPs intend to vote against the agreement. "Any other deal really only exists in people's imagination".
"I will not be blackmailed into voting for the Prime Minister's deal when there are clearly other options available. It would open the door to more division and more uncertainty, with all the risks that will entail", she said.
The European Parliament has investigated the clause in the treaty and concluded, as did the Scottish courts that took up the legal challenge, that only the ECJ can make a legal ruling on the matter.
Looking towards December 11, May said there is a choice the House of Commons will have to make.
"It is no surprise some are already trying to lay down markers again for the future relationship, but they should be getting used to the answer by now: it is not going to happen".
May's Conservative Party holds 316 seats in the House of Commons, relying on support from the 10 Northern Ireland Democratic Unionist Party (DUP) MPs to give her a potential of 326 votes, a majority of one.
Downing Street has said in the "unlikely event" that Parliament could not agree on the terms of withdrawal, "all necessary action" would be taken to prepare the United Kingdom to leave without an agreement.
The prime minister subsequently confirmed the date to MPs, saying that she was "looking ahead to [the vote on] December 11".