The Tory leader told the BBC on Friday he was "always grateful for advice" but he would not enter into election pacts.
His comments come after the US President said Farage and Johnson would be "an unstoppable force".
Downing Street sources say there are no circumstances in which the Tories would work with the Brexit Party.
In an interview with the BBC, the Prime Minister said the "difficulty" of doing deals with "any other party" was that it "simply risks putting Jeremy Corbyn into Number 10".
"The problem with that is that his (Corbyn's) plan for Brexit is basically yet more dither and delay," Johnson said.
Johnson also said there was "no question of negotiating on the NHS" as part of any future trade deal with the US, but he did not rule out expanding the amount of private provision in the health service in the future.
But Labour's shadow health secretary, Jonathan Ashworth, said the public "can't trust the Tories on the NHS", saying they would "increase privatisation even further and do a deal with Donald Trump".
When pushed on whether he would rule out a deal with Farage, Johnson replied: "I want to be very, very clear that voting for any other party than this government, this Conservative government... is basically tantamount to putting Jeremy Corbyn in."
The UK is going to the polls on December 12 following a further delay to the UK's departure from the EU, to January 31, 2020.
( With inputs from IANS )