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Morgan is making his UK television return on Monday for his show on new channel TalkTV and has trailed an interview with Trump for the first episode.
In a clip for the interview, Trump can be seen getting up off his chair and seemingly storming off while asking for the cameras to be turned off.
The front page of Thursday’s The Sun splashed on the apparent walkout – but Farage has described both the headline and the trail for the interview as “fake news”.
Trump himself has also said the Morgan promo was “doctored”.
In a video posted to his YouTube channel, Farage recalled telling the Trump that Morgan had “gone around boasting in the UK media” that he was his friend while also “stitching you up”.
He claimed to have told Trump: “If you do this interview, here is the truth,” before handing him the dossier.
Farage added: “Trump clearly was extremely displeased. The interview, because of that, wasn’t quite the friendliest it ever could have been.”
Watch: 'I'm a more honest man than you': Trump appears to storm out of Piers Morgan interview
Hitting out at claims Trump stormed out of the interview, Farage continued: “The interview had been concluded, it was finished, it was over.
“They said their ‘thank yous’ and Trump got up for his next appointment.
“Trump has been completely and utterly stitched up by Piers Morgan.”
Responding to Farage, Morgan tweeted: “Actually, Nigel, the only stitching up was done by you when you sent him that dossier of my criticisms (which I stand by) in a pathetic attempt to ruin my Trump interview.”
Morgan later tweeted Trump's director of communications, saying: "Hi Taylor, the fact you weren’t able to stop the interview, and the fact President Trump had to be told about my previous public criticisms of him by some dodgy bloke in London rather than by you suggests he urgently needs a new director of communications! I’d pack your bags, mate."
Writing in The Sun, Morgan said Trump was “foaming at the mouth” ahead of the interview after being handed a three-page document detailing critical comments Morgan had made about him on topics including his “woeful handling” of the coronavirus pandemic and his response to defeat in the 2020 presidential election.
Morgan added: “I’d never seen him so livid or felt so uncomfortable in his presence as I did right now.
"He was almost foaming at the mouth and kept shaking his head slowly and menacingly at me, like Don Corleone when he felt he was disrespected.
“There was no point in trying to deny the quotes. I’d said them, and I’d meant them.”
Trump subsequently released a statement saying Morgan had “attempted to unlawfully and deceptively edit his long and tedious interview with me” in order to make it look like he had walked out.
He added: “The good news is that the interview was taped by us as a means of keeping him honest.”
Trump’s son, Donald Trump Jr, waded into the row, revealing that audio from the end of the interview painted a different picture to what Morgan has painted.
He added: “It’s totally brutal how misleading and deceptive the Morgan promo edit is.”
Trump's director of communications Taylor Budowich added: "This is a pathetic attempt to use president Trump as a way to revive the career of a failed television host.”
The trailer for the interview shows Morgan asking Trump whether he still believes the 2020 election was stolen.
The pair are seen arguing and speaking over one another, while Trump calls Morgan a "fool" for saying he lost the election fairly.
Piers Morgan Uncensored launches on TalkTV, a subsidiary of Rupert Murdoch's News Corporation, at 8pm on Monday.