"I don't wish to cast any aspersions on the president, but in that respect he is patently in error," Johnson said. "Anybody who looks at our deal can see it is a great deal... it allows us to have full unfettered control of our tariff schedule".
Trump on Thursday told friend Farage - whose Brexit Party will be up against Johnson in the December 12 election - that "under certain aspects of the deal - you can't do it, you can't trade."
Farage said Sunday he will not seek to become an MP in next month's vote, but that his party would challenge the Tories across the country.
"Do I find a seat, try and get myself into parliament? Or do I serve the cause better traversing the length and breadth of the United Kingdom supporting 600 candidates - I've decided that the last course is the right one," he told BBC's Andrew Marr.
The Brexit Party was formed earlier this year to fight the European parliamentary elections, where it emerged as the runaway winner.
But Farage, who has been a member of the European Parliament since 1999, has already tried to become an MP on seven occasions, losing out each time.
Farage has been a harsh critic of Johnson's deal but has called for an election pact with the Conservatives, even as the prime minister ruled out such a deal. "I ruled out a pact with everybody because I don't think that it's sensible," said Johnson.
"What I can say respectfully to the leaders of all other parties is, alas, the only likely consequence of voting for them, rather than for us as Conservatives is that you're making it more likely that you will thereby get (Labour leader) Jeremy Corbyn".
With the Tories relying on Brexit supporters to gain a majority on December 12, a strong showing by Farage's party threatens to split the leave vote and open the door for the Labour party. But Farage kept up his opposition to Johnson's deal, carrying a two-page advert in the Sunday Express which said that "Boris' deal is not Brexit".
People "need to understand that, actually, what's on offer is a close linkage with all the European institutions, a next negotiating phase of at least three years," he told Marr.
Johnson had vowed he would rather be "dead in a ditch" than tolerate another extension to the tortuous Brexit process, which began in 2016 after a knife-edge public referendum.
On Sunday, he blamed Britain's failure to leave as scheduled on October 31 on parliament passing what he called the "Surrender Act". This demanded that the government ask for, and accept, a delay from the European Union rather than leave without the deal being ratified.
MPs gave provisional support for his deal, but derailed Johnson's plans to get the bill through parliament before the deadline day, leading him to push for the December 12 vote to break the impasse.