Theresa May has hailed the "powerful vote of confidence" in Britain Donald Trump and other world leaders have shown with their "strong desire" to strike new trade deals after Brexit.
The Prime Minister said she is "optimistic and positive" about a future pact with the US after the president said he believed an agreement could be reached "very, very quickly".
Following talks on the margins of the G20 summit in Hamburg, Mr Trump hailed the "very special relationship" he had developed with the PM.
He said he expected an agreement on new trading arrangements with Britain to be "very powerful".
Mrs May said: "We are optimistic and positive about this."
As well as talks with President Trump, Mrs May has met the premiers of China, Japan and India for one-to-one meetings during the two-day summit.
She said: "I've held a number of meetings with other world leaders at this summit and have been struck by their strong desire to forge ambitious new bilateral trading relationships with the UK after Brexit.
"This is a powerful vote of confidence in British goods, British services, Britain's economy and the British people and we look forward to building on these conversations in the months ahead."
Mrs May insisted she was confident the UK would also secure a good deal with the EU "because it's not just about what's in the interests of the United Kingdom, it's about what's in the interests of the remaining 27 members states in the European Union and I think it is in the interests of both sides to have that good trade agreement".
She added: "But I'm also optimistic about the opportunities that we will see in the rest of the world.
"Some of the countries I have been talking to here who have shown great interest in working with us on trade arrangements in the future, the United States, Japan, China, India, these are all huge world economies.
"This is an important development for the United Kingdom and I look forward to developing those trade deals as well."
Mr Trump and the Prime Minister agreed in a 50-minute meeting to prioritise work on the trade deal so it can be completed "as soon as possible" after Brexit, a UK Government official said.
They also pledged to look at areas where trade can be deepened before Britain quits the EU. The discussions ran 20 minutes over schedule and the leaders continued to chat as they walked to another meeting at the Hamburg summit.
Mrs May also held a 20-minute meeting with her Japanese counterpart Shinzo Abe, which focused on trade and North Korea's nuclear missile programme.
Japan's new trade deal with the EU, signed off on Thursday, "could form the basis" of an agreement between London and Tokyo following Brexit, Mrs May told the Japanese premier.
Indian prime minister Narendra Modi told Mrs May he wanted to see economic links with the UK deepen now and after Brexit and they agreed to work together to put a "concrete" plan in place, according to a UK government official.
Mrs May also held talks with President Xi on Friday over a future deal with China.
President Trump said: "There is no country that could possibly be closer than our countries.
"We have been working on a trade deal which will be a very, very big deal, a very powerful deal, great for both countries and I think we will have that done very, very quickly."
He added: "Prime Minister May and I have developed a very special relationship and I think trade will be a very big factor between our two countries."