Mr Trump later said he and Mr Johnson had shared a "great discussion" in which they talked about moving "rapidly on a US-UK free trade deal" after Britain's departure from the bloc.
But the European Commission, the EU executive which has led Brexit negotiations with London, dismissed the proposal in Mr Johnson's letter that the backstop could be replaced with a "commitment" to find "alternative arrangements".
Similarly, it said that "the letter's suggestion that two separate legal, political, economic and monetary jurisdictions already exist on the island and can be managed with an open border is misleading".
A spokeswoman said of the letter that, "it does not set out what any alternative arrangements could be" and "recognises that there is no guarantee such arrangements would be in place by the end of the transitional period".
"The backstop is insurance to avoid a hard border on the Irish island unless and until an alternative is found", Tusk said.
Britain's Prime Minister Boris Johnson wrote that time is short and proposed that the backstop should be replaced with a commitment to put in place (alternative) arrangements as far as possible before the end of the transition period, as part of the future relationship. "Even if they do not admit it", Tusk wrote on Twitter.
Johnson is due to meet the German chancellor, Angela Merkel, on Wednesday and the French president, Emmanuel Macron, on Thursday, so there will be more of an idea how his ideas have gone down with major European Union leaders then.
The UK does not want any border - effective or virtual - between Britain and Northern Ireland.
"I mean, when you look across unionism, we are all against the backstop so that should concern the Dublin Government, because if the Dublin Government is genuine and values the Belfast Agreement they should be very concerned that unionism does not support the backstop".
"If we had to put odds, there is a 40% chance of a deal that passes through given the flexibility shown by the United Kingdom government, a 40% chance of a hard Brexit and a 20% chance of continuing as is".
BREXIT BRINKMANSHIP European diplomats expect little progress on Brexit until the British domestic landscape becomes clearer when parliament returns on September 3 - after which the opposition Labour Party has vowed to try to collapse Johnson's government.
"It is rank hypocrisy for Boris Johnson to claim to be acting in the interests of the peace process, claiming it will be damaged by the backstop", she said.
It is not the first time Mr Trump has spoken of the potential for a trade agreement after Britain leaves the European Union, but came as a senior USA politician warned that politicians could block a future deal if the Good Friday Agreement is undermined.