The Republican, who beat Democratic rival Hillary Clinton in the election earlier this month, is due to take office on 20 January.
Earlier this year, MPs debated the idea of banning Mr Trump from the UK.
But the government has stressed the importance of maintaining the "special relationship" between the countries.
Prime Minister Theresa May's official spokeswoman said: "An invitation for a state visit is one of the things that is under consideration following the election of a new US president. One of the issues under consideration is the 2017 state visits."
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State visits are formal trips to the UK by foreign heads of state from overseas. There are normally two a year and invitations are sent on the advice of the Foreign Office.
Recent state visits to UK
November 2016 - President Juan Manuel Santos of Colombia
October 2015 - President Xi Jinping of China
March 2015 - President Enrique Pena Nieto of Mexico
October 2014 - President Tony Tan Keng Yam of Singapore
April 2014 - President Michael D. Higgins of Ireland
November 2013 - President Park Geun-hye of South Korea
The Queen acts as host, with the visitor usually staying either at Buckingham Palace or Windsor Castle.
Visits normally begin with a ceremonial welcome attended by The Queen and other senior members of the Royal Family, while a State Banquet is held in his or her honour.
Mr Trump's predecessor, Barack Obama, made a state visit to the UK in 2011.
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