Theresa May said several G20 leaders displayed a "strong desire" to strike mutually beneficial trade arrangements with the UK after Brexit.
She hailed the potential for "ambitious" deals after Donald Trump said Britain would "thrive" outside the EU.
Mrs May said world leaders also displayed a "strong desire" to strike mutually beneficial deals with the UK during talks in Hamburg.
Japan, China and India all expressed and interest in developing their trading relationships with Britain.
Mr Trump said he expects a US trade deal with the UK to be completed "very, very quickly" and that it would be a "very powerful deal".
The two leaders have also pledged to look at areas where trade can be deepened before Britain quits the EU.
However, despite Mrs May saying she was "dismayed" by Mr Trump's withdrawal of the US from the Paris climate change agreement, she failed to raise the topic during their bilateral meeting.
A UK government official said Mrs May and Mr Trump spent a "significant" amount of time discussing trade and that the talks were entirely "positive".
The US president hailed the "very special relationship" he had developed with Mrs May, saying: "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."
Later, she maintained the UK was looking to forge comprehensive trade deals with "old friends and new partners".
"I have 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," she said.
"This is a powerful vote of confidence in British goods, British services, Britain's economy and the British people."
Ahead of their 50-minute meeting, President Trump insisted he still plans to visit Britain.
It follows speculation that a planned state visit to the UK may not go ahead this year and the likelihood any visit would be met with protests.
Asked when he will be visiting, he replied: "We'll work that out."
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."
Mrs May said she was looking forward to welcoming President Trump to the UK, but specified that both sides were still working on what an "appropriate" date would be.
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 Prime Minister.