Donald Trump's controversial first visit to the UK as US President is set to take place later this year.
The two leaders asked officials to work together on finalising the details of the trip during talks at the World Economic Forum in Swiss ski resort Davos.
The development comes just weeks after Mr Trump cancelled a proposed trip to officially open the US Embassy in London.
"The PM and President concluded by asking officials to work together on finalising the details of a visit by the President to the UK later this year," Downing Street said.
During their talks Mrs May also raised the issue of aircraft manufacturer Bombardier which has a major plant in Northern Ireland and is at the centre of a US trade dispute.
The United States Commerce Department has confirmed plans to impose duties of 292% on imports of Bombardier's C Series commercial jets after American rival Boeing claimed the firm had been given unfair state subsidies by the UK and Canada.
A Downing Street spokesman said: "The two leaders began by discussing Bombardier, with the PM reiterating the importance of the company's jobs in Northern Ireland."
Any visit by Mr Trump is expected to attract protests and concerns about the level of opposition is thought to have been one of the reasons why it has taken so long for a trip to take place.
Mrs May extended an invitation for a state visit - officially on behalf of the Queen - when she became the first foreign leader to visit the Trump White House in January 2017.
The planned trip to open the embassy would have been a lower-key official visit but Mr Trump cancelled it, publicly blaming a poor financial deal for the property which he said was in an "off location" south of the Thames - although allies indicated that possible protests could have been a factor.
Brexit has heightened the importance of the UK-US "special relationship", with the prospect of a transatlantic trade deal a major goal for Mrs May.
Mr Trump said he expects a "tremendous" increase in trade with the UK as a result of talks which he said were getting under way.
The US President sought to kill off suggestions of a rift with Mrs May sparked by her rebuke over his retweet of videos posted by a UK far-right group and the cancelled embassy visit.
Declaring that he wanted to correct the "false rumour" of a transatlantic falling-out, he assured Mrs May: "We love your country."
Britain and the US are "joined at the hip" militarily and America will always be ready to "fight for you", he told her.
Mrs May said they had enjoyed a "great discussion" about the prospects for improved trade relations.
And Mr Trump said he expected trade between the US and UK to "increase many times" over the coming years.
"The trade concepts and discussions that will be taking place are going to lead to tremendous increases in trade between our two countries, which is great for both in terms of jobs," said the president.
"We look forward to that and we are starting that process pretty much as we speak."
Mrs May said she and the president enjoyed "a great discussion" in their meeting, adding: "We continue to have that really special relationship between the UK and the United States, standing shoulder to shoulder because we are facing the same challenges across the world."
A "good trade relationship in the future" will "be to both our benefits", she said.