Farage, the former and now interim leader of the U.K. Independence Party, met with Trump in NY on Saturday at Trump Tower, suggesting he could act as a go-between to help smooth British-American relations.
Farage, who appeared alongside Trump to endorse the Republic candidate at a rally in the southern state of Mississipi prior to the election, had a meeting with the businessman-turned-politician at Trump's NY tower on Saturday.
Mrs May's official spokeswoman said the government already has "well-established" channels of communication with Mr Trump's team.
Farage is the first foreign politician to meet with Trump since the USA presidential election, with British Prime Minister Theresa May not due to follow suit until early next year.
"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-US President Ronald Reagan in the 1980s.
Chuka Umunna said that the United Kingdom should take advantage of that relationship, but dismissed the notion that Mr Farage is now central to our dealings with Mr Trump.
Banks, who was one of five people in Farage's entourage entertained at Trump Tower in Manhattan this weekend, said the hour-long meeting between Trump and Farage on Saturday was serious rather than lighthearted and touched on serious issues from Brexit to North Atlantic Treaty Organisation.
Nigel Farage has said the way Number 10 has rejected his offer to act as a Trump administration go-between is "astonishing".
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The shock outcome of the USA election prompted many commentators to draw parallels with Britain's June 23 referendum after Trump said the vote would be "Brexit plus plus plus".
Professor Rob Ford, an expert in the radical right at Britain's University of Manchester, said Farage could well be a useful conduit into a Trump White House.
After spending more than an hour with the president-elect, the interim Ukip's leader insisted Theresa May should stop running him down and instead use his closeness to the tycoon-turned-next U.S. head of state to "put the national interest first".
"I'm confident he will be a good president", Farage added.
"He said he had a nice conversation, although some of his team had reservations about what members of the Cabinet have said during the election".
A photo of the two men - Farage grinning broadly and Trump giving a thumbs up - was widely published in Britain, with one headline reading: "The victory of the outsiders".
"Whatever No 10 say, Nigel has a hotline to Trump and some of his closest advisers".