Trump caused diplomatic ripples when he said on Twitter that Farage, interim leader of the anti-immigrant UK Independence Party (UKIP), would "do a great job" in the post.
A spokesman for the office of British Prime Minister Theresa May blocked the idea, telling CNN the UK appoints its own ambassadors and that the position was filled.
Trump's expression of a preference for UK ambassador to Washington was a startling break with diplomatic protocol. It is unheard of in recent years for any US official to make such a suggestion to an ally.
It was particularly awkward for the UK as Farage has long been a thorn in the side of May's Conservative party. UKIP's political surge forced her predecessor, David Cameron, into holding a referendum on Britain's membership of the European Union. That ended in the shock vote to leave the EU and Cameron's political downfall.
Britain put its best diplomatic face on Trump's unexpected intervention. "There is no vacancy. We already have an excellent ambassador to the US," the Downing Street spokesman said.
British Foreign Secretary Boris Johnson echoed the comments in parliament Tuesday.
"We have a first-rate ambassador in Washington doing a very good job of relating both with the present administration and with the administration to be, and there is no vacancy for that position."
Kim Darroch took up the role of US ambassador in January, with three decades of diplomatic experience under his belt. His career has focused on national security and European Union policy issues.
Farage: 'The world has changed'
Farage responded to Trump's public endorsement Tuesday, saying on Twitter that he was "in a good position" to help the bilateral relationship.
In an article for Breitbart News, formerly run by Trump's chief strategist Steve Bannon, Farage said he was surprised by Trump's tweet. It was a "bolt from the blue," he wrote, and called for the British government to take note. "The world has changed and it's time that Downing Street did too," he said.
Farage told CNN's Richard Quest after Trump's election that he doubted he would be offered the role of ambassador, given his relationship with the governing party in the UK.
"The British Conservative Party are incredibly snobby about me. They've always looked down upon me. They find it very difficult to even have a conversation with me," he said.
"If they came to me and said, 'Would you want to do something that would help the benefit of our country, and our trade relationship with America, and that would work both ways,' if they came and asked me to do it, of course I'd think very hard about doing it. But I honestly think it's unlikely."
Bonding over Brexit
Trump and Farage have often expressed admiration for each other. Trump often compared his campaign for the presidency to Farage's efforts in the UK. In August, Trump tweeted: "They will soon be calling me MR. BREXIT!" and on Election Day he told supporters that his policies would be "Brexit plus plus plus."
Farage was quick to board a plane after Trump's victory, congratulating him three days later during a meeting at Trump Tower in New York.
On Twitter, Farage posted a photo of the pair, with Trump giving his signature thumbs-up gesture, and Farage wide-mouthed with excitement.
"I'm confident he will be a good President. His support for the US-UK relationship is very strong. This is a man with whom we can do business," Farage posted.
UKIP facing investigation
Back in the UK, Nigel Farage's UKIP is facing scrutiny over whether it breached funding rules.
The UK Electoral Commission has opened an investigation focused on financial grants to the party.
In a statement, the commission said grant funding from the European Union can be used by a political party to help pay for things like campaign costs connected to European elections -- but cannot be used to fund national parties, election candidates or a range of other purposes.
The commission added that after receiving information from the European Parliament it will look into whether UKIP broke UK election law.