An early visit to the UK from Trump would also allow the Prime Minister to swat away claims that Nigel Farage has closer links to the controversial tycoon, which have been dismissed by Downing Street.
It would also provide a potential opportunity to warm up the apparently anti-global trade president-elect over the idea of a post-Brexit trade deal.
A Downing Street spokeswoman said: “No visit has been organised but the Prime Minister is looking forward to welcoming the president-elect to the UK when he chooses to visit.”
The PM has been invited to visit Trump at the earliest possible opportunity and she is expected to travel to the US after he is inaugurated on January 20.
But a state visit by Trump to the UK would allow her to roll out the red carpet and offer a meeting with the Queen.
A Buckingham Palace spokesman said: “State visits and other meetings with overseas heads of state are organised on the advice of the Government.”
It came as Mrs May faced fresh calls to deliver what will be seen as a “hard Brexit” from a group of 60 Conservative MPs including prominent former cabinet ministers.
Cameron-era ministers Michael Gove, Iain Duncan Smith, John Whittingdale and Theresa Villiers urged the PM to pull Britain out of the European single market and the customs union.
They claimed getting out of the single market free trade zone was crucial for the UK to become free of Brussels regulations.
Pulling out of the customs union, which sets common tariffs for goods from countries outside it, would be the only way to strike trade deals with other nations, they said.
Eleven Labour, DUP and Ukip MPs also reportedly backed the call.
A Government spokeswoman insisted there were “no binary choices” in the UK’s future trading relationship with the EU and that the Government would pursue a bespoke deal rather than an “off the shelf” solution.
She added: “That’s why the Government is painstakingly analysing the challenges and opportunities for all the different sectors of our economy.
“The Prime Minister has been clear that she wants UK companies to have the maximum freedom to trade with and operate in the single market – and to let European businesses do the same here.”
Shadow chancellor John McDonnell accused the Tories of pursuing a “Trump-lite” hard Brexit “with a minuscule increase in infrastructure investment, a pernicious immigration policy, the destruction of workplace rights and environmental protections, alongside a race to the bottom in taxes for the rich and in wages for the rest of us”.
He also defended his speech earlier this week in which he committed Labour to not blocking or delaying Brexit, saying the party has long advocated changes to single market regulations such as state aid rules and “enforced deregulation and privatisation”.
Writing in The Observer, McDonnell said: “As a democrat, I respect the referendum result.
“Not to do so would sow even deeper divisions within our society. But that does not mean we cannot use the period of the Brexit negotiations positively to work with our socialist, social democrat and progressive friends in Europe to construct a new relationship based upon solidarity, co-operation and democracy.”