The PA news agency understands Ms Truss questioned whether that position is tenable in the long term and has considered other routes to improving trading ties with the States.
She was to meet with the US president alongside Prime Minister Boris Johnson at the White House on Tuesday.
But, after a return to New York for further talks relating to the UN General Assembly, she will head to Mexico City to open a new British embassy.
She was understood to be considering the possibility of the UK joining the United States-Mexico-Canada Agreement.
A diplomatic source said: “There are a variety of different ways to do this. The question is whether the US administration is ready.
“The ball is in the US’s court. It takes two to tango.”
A comprehensive free trade agreement with the US was touted as one of the prizes of Brexit during the 2016 referendum.
But Mr Johnson has expressed doubts that he will be able to get such a deal in place ahead of the next general election, saying Mr Biden has “a lot of fish to fry”.
Asked if he would get the deal by 2024, the Prime Minister told Sky News: “We will keep going with free trade deals around the world including in the United States.
“I have plenty of reason to be optimistic about that. But the Americans do negotiate very hard.”
