Nigel Farage has expressed a willingness to serve Donald Trump “formally or informally” in the pursuit of improved US-UK relations.

“I just happen to know a few people in his administration,” the former UK Independence party leader told CNN on Saturday. “I’ve clearly got Trump’s confidence and I want us to move as quickly as we can towards a free trade deal between the UK and the USA.”

Trump infamously tweeted that Farage should be British ambassador to the US. Asked if he envisaged taking a formal diplomatic role, Farage said: “Do you know, I honestly don’t know the answer to that.

“But what I will say is I would like, formally or informally, to do whatever I can to bring our great nations a bit closer together.”

Farage was the first British politician to meet Trump after the New York billionaire’s victory in the presidential election, attending Trump Tower in Manhattan for a talk which reportedly included “freedom and winning” and Trump’s supposed willingness to reinstall a bust of Churchill in the Oval Office.

Theresa May, in contrast, was only the 11th foreign leader called by Trump as president-elect. Trump reportedly told the British prime minister: “If you travel to the US you should let me know.”

Trump and May spoke for a second time this week, Downing Street said, agreeing on the “importance of Nato”.

May’s government firmly rejected Trump’s suggestion that Farage should be ambassador, prompting Ukip donor Aaron Banks to say the prime minister would be “nuts” to reject the help of someone with “a hotline to Trump and some of his closest advisers”.

Banks, who also attended the Trump Tower meeting, also said Farage enjoyed a close relationship with Steve Bannon, the controversial senior Trump aide who once ran the far-right website Breitbart.com.

“I do think the special relationship [between the US and the UK] is very important,” Farage said on Saturday from Washington, where he was reportedly due to meet members of Trump’s transition team.

“It was significantly devalued in Obama’s time; he looked to [Angela] Merkel, he looked to the European Union and not to us as an independent country.

“So post-Brexit we’ve got a chance to start all over again with a president in Trump who is anglophile, he is pro-British, he knows the things we’ve shared together over the years, the good and the bad, the tough times as well as the good.”

Though touting his proposed “free trade deal” between the US and UK, Farage did not address the success in the rust belt states of Trump’s explicitly protectionist “America First” campaign message.

A deal, he said, would be “good for both of us and it will also send a signal to the European Union that there’s a bigger world outside Europe and that Britain can manage just nicely”.

In another shot at Obama, referring to remarks by the US president before the Brexit vote about the possible trade consequences of Britain leaving Europe, Farage said: “No longer do we have a president who says that we’re at the back of the line.”

He also said Hillary Clinton, Trump’s opponent, had “wanted the European Union to be a prototype for a bigger model across the whole world”. Instead, Farage predicted, “this European Union is dying before your eyes”.

“I can’t tell you how long it will take,” he said, “but basically it’s finished.”

Farage, who was introduced by CNN host Michael Smerconish as a “foreign leader”, is no longer the leader of Ukip. In the Brexit referendum, he played a second-fiddle role to two prominent Conservatives: Boris Johnson, now foreign secretary, and the former justice minister Michael Gove.

Farage is a member of the European Parliament but has failed in seven attempts to be elected as an MP in Britain. He recently said he might try for an eighth time.

He has also reportedly said he might move to the US, and that he expects to be invited to Trump’s inauguration, in Washington on 20 January.