The interim Ukip leader, who was the first British politician to meet the controversial tycoon since his election success, said “petty personal differences” should be set aside.
Number 10 dismissed suggestions that the Ukip leader might become the “third person” in the relationship between Mr Trump and the Prime Minister, insisting that the Government already has “well-established” channels of communication.
But Mr Farage told ITV’s Good Morning Britain: “Are they suggesting that they sack all civil servants, they sack all ministers and the Prime Minister does everything? It’s nonsense, isn’t it?
“This Government has got a massive job to do with Brexit .... all I have said is ‘I’ve got some connections, I’d like to help’.”
Mr Farage, who claimed he had been “bombarded” with calls from businesses seeking his help to strike up a relationship with the Trump administration, said: “If Government policy is to secure a free trade deal with the USA, if I can help in any way make that happen I will do so.
“If they don’t want me, that’s fine.”
He dismissed reports that Mr Trump’s team would talk to him ahead of the British Government, but added: “Commonsense says that I should be useful some how to this country in getting that relationship with America back to where it should be.
“We’ve had eight years of (Barack) Obama, who didn’t like us. Trump likes this country.”
The interim Ukip leader said he had made “a couple of bob” as a result of betting on victory for Mr Trump: “I won enough money to buy myself a couple of pints every day for the next year.”
Meanwhile, Mr Trump was continuing to piece together the administration that will lead the USA from January 20 when he takes office.
Former New York mayor Rudy Giuliani has emerged as the favourite to serve as secretary of state.
John Bolton, a former US ambassador to the United Nations, is also believed to be in the running for the position.
Mrs May and Foreign Secretary Boris Johnson briefed Cabinet at its regular weekly meeting in Downing Street on their conversations last week with Mr Trump and Vice President-elect Mike Pence.
Mrs May and Mr Johnson said they would continue their engagement and build on links with the Trump team established by the UK Embassy in Washington, said the PM’s spokeswoman.
The PM said that the UK will want to work with the incoming Trump administration particularly on the importance of Nato, international efforts to bring peace to Syria and standing up for free trade.
Asked if Mr Farage was mentioned during the five-minute exchange around the Cabinet table, the spokeswoman said: “No, it was a discussion about the Government’s engagement with the US.”