The US President-elect tweeted that Mr Farage would "do a great job" as Britain's ambassador to the US
Britain's ambassador to the US has met Theresa May for talks, after the British Prime Minister rejected Donald Trump's call for him to be replaced by Nigel Farage.
Downing Street did not comment on the contents of Nigel Kim Darroch's conversation with Mrs May, but made it clear Mrs May regards him as "an excellent ambassador who is doing a fantastic job".
The US President-elect had earlier suggested Nigel Farage should be appointed as Britain's man in Washington, saying he would do "a great job".
The unusual intervention came in a late-night tweet.
Many people would like to see @Nigel_Farage represent Great Britain as their Ambassador to the United States. He would do a great job!— Donald J. Trump (@realDonaldTrump) November 22, 2016
Diplomat Nigel Kim Darroch, who was appointed to the position earlier this year, caused controversy following Mr Trump's election victory after a document leaked to the Sunday Times outlined his view that the President-elect was “open to outside influence.”
Speaking to Sky News after Mr Trump's tweet, Mr Farage said he was "flattered" and when asked if he was suited to the ambassadorial role, he replied: "The world has changed."
"I would like to play a positive role in relations between our two countries," he said.
Writing in a blog, Mr Farage said Mr Trump's tweet was like a "bolt out of the blue" and hit out at Downing Street.
He said: "I have known several of the Trump team for years and I am in a good position with the President-elect's support to help. The world has changed and its time that Downing Street did too."
However, a Number 10 spokesman said there was "no vacancy for an ambassador to the United States".
The UK's Foreign Secretary Boris Johnson said: "We have a first rate ambassador in Washington doing a very good job.
"Britain and America have an enduring and strong special relationship.
"And as the Prime Minister said during her call with President-elect Trump on the 10 November, we look forward to working with his administration to ensure the security and the prosperity of both our countries and the world in the years ahead."
Mr Farage has previously expressed an interest in being Mr Trump's ambassador to Europe and offered to act as an intermediary between the Trump administration and the UK government.
The suggestion of the acting UKIP leader acting as a "go between" was rejected by Downing Street, who said he would not be the third person in the relationship between Mrs May and Mr Trump.
Mr Farage travelled to the US to campaign alongside Mr Trump in Mississippi ahead of the presidential election and became the first British politician to meet the billionaire after his victory.
The two met in New York just days after Mr Trump became President-elect.
In an interview with Sky News after the meeting, Mr Farage rounded on the Government for rejecting suggestions he could help cement US/UK relations.
And he called for the current ambassador to resign because he is a "fanatical Europhile and part of the old regime" whose views would be "diametrically opposed to Trump's".
Mr Trump is in the process of appointing his team and on Monday released a video updating Americans on the White House transition in which he said on his first day in office he would withdraw from the Trans-Pacific Partnership.