Mr Trump has not visited the UK since he was elected to the White House in November 2016, with a potential visit to London to open the US Embassy in January not taking place.
The Daily Mail report a visit from Mr Trump has been “pencilled in” for July, while the Daily Telegraph quoted an unnamed US official saying the president was looking at coming in the “late summer”.
Previously Ministers have discussed plans for Mr Trump to meet the Queen at Balmoral Castle before visiting one of his two golf courses in Aberdeenshire and Ayrshire.
Mr Trump cancelled a visit to the UK to open the new American embassy, criticising its move from Grosvenor Square in the prestigious Mayfair district of central London to an “off location” at Nine Elms, south of the Thames.
The president blamed the cost of the new embassy and its location south of the River Thames, saying it was a “bad deal”.
His cancellation prompted media speculation that reasons for the snub included that Mr Trump had been offended by perceived slights against him by UK public figures.
Foreign Secretary Boris Johnson accused London Mayor Sadiq Khan of endangering the so-called “special relationship” after he said president had “got the message” from Londoners and would have been met by “mass peaceful protests” if he went ahead with the visit.
The president has endured a turbulent relationship with Mrs May since taking office, with the Prime Minister publicly criticising statements he has made on Muslims, terrorism and climate change.
However Mr Trump said he was grateful for the support from the UK and France when a coalition of the three countries conducted targeted air strikes in Syria last weekend.