LONDON (Reuters) - Britain has established ways of contacting President-elect Donald Trump and is building contacts with him, a spokeswoman for Prime Minister Theresa May said on Monday when asked about a U.S. trip by Brexit campaigner Nigel Farage.

Farage met Trump on Saturday, the first British politician to visit the newly elected president of the United States, and suggested that he should become a "go-between" for the government and the U.S. administration.

"We have established routes of engagement with the President-elect and his team. Our diplomatic staff have been building those contacts and links in the run up to the election," the spokeswoman said when asked whether the government would accept advice from Farage.

Repeatedly questioned about a possible role for Farage, the spokeswoman referred to a call between May and Trump on Thursday, in which Trump invited the British Prime Minister to visit him as soon as possible and underlined the importance of the U.S.-British relationship.

"The President-elect talked about enjoying the same close relationship that Reagan and Thatcher did," the spokeswoman said referring to the close ties developed between ex-prime minister Thatcher and former-U.S. President Ronald Reagan in the 1980s.

"I don't remember there being a third person in that relationship," she said.

(Reporting by William James, writing by Elizabeth Piper; editing by Guy Faulconbridge)