Following a meeting in the White House on Tuesday, Mr Raab praised the president's "warmth and enthusiasm" for the UK during his first visit to the United States since assuming office last month.
He said: "We talked about all the things that we want to do together in the post-Brexit vision for the UK, whether it's getting a free trade deal done, and there's obviously going to be a lot of work to make that happen.
"There's a huge appetite on both sides to achieve that."
The Foreign Office said Mr Raab met the president and Vice President Mike Pence in Washington on the second leg of his tour of North America.
On meeting the president, Mr Raab added: "We had a preliminary chat, he called me into the Oval Office. He was effusive in his warmth for the United Kingdom.
"It's amazing to hear an American president talk about our country in such warm terms."
Earlier, in a Twitter posting, Mr Raab said that he had held “wide-ranging talks” with Mr Pence on a series of issues including China, Iran and a post-Brexit trade deal.
No further details of the meeting with Mr Trump were immediately available.
However, the president has already expressed his enthusiasm for the appointment of Boris Johnson as Prime Minister after relations with Theresa May soured during her final days in office.
Mr Raab is due to have further discussions on Wednesday with other senior administration figures including Secretary of State Mike Pompeo and National Security Adviser John Bolton.
He flew into the US capital following talks on Tuesday in Toronto with Canadian foreign minister Chrystia Freeland and is due to travel on to Mexico following his meetings in Washington.