Mr Farage insisted he could be a “bridge” between London and Washing and described Downing Street’s position as “small-minded”.
But Liberal Democrat leader Tim Farron criticised Mr Farage for continuing to pursue his ambition to represent UK interests.
He said: ““Nigel Farage has absolutely no right to speak for Britain on the world stage.
“The vast majority of British people do not share his hateful vision and want to keep our country open, tolerant and united.
“Someone who has spent their career trying to divide our country is not fit to represent it abroad.”
Mr Farage campaigned for Mr Trump during the presidential race and was the first UK politician to meet him following the election.
Mr Trump later suggested Mr Farage could do a “great job” as British ambassador to the US.
Mr Farage said: “If you wanted to get business from the biggest company in the world and there was someone there that had the contacts and connections, the first thing you do is to employ that person.
“I find the fact that Downing Street don’t want to acknowledge me, have banned cabinet ministers from even talking to me, it shows you the sort of very small-minded, petty, tribal level that British politics is at - it’s a shame.”
Quizzed on his relationship with the president-elect and what he would compare it to, he said: “I can’t make big decisions, but I tell you what I can do, what I genuinely can do.
“Is I can help to be a bridge between the government in this country and the new trade department in this country and not just Donald Trump, but his team and his administration.
“And I would like to do that.”