The Prime Minister is said to be reluctant to be interviewed by Mr Neil after he grilled Jeremy Corbyn over anti-Semitism in the Labour Party on Tuesday.
Mr Johnson offered to appear on Mr Marr’s programme on Sunday, but the BBC turned him down because he would not agree to an interview with Mr Neil, The Daily Telegraph reported.
LBC radio’s Nick Ferrari asked Mr Johnson on Friday if he would be interviewed by Mr Neil, but the Prime Minister dodged the question.
Mr Johnson replied: “What people really want to hear is, what are we doing to take the UK forward?”
In Mr Corbyn’s interview with Mr Neil, the Labour leader refused four times to apologise for his handling of anti-Semitism allegations.
The issue then dominated the news cycle the next day.
A BBC source told The Daily Telegraph: “He won’t be doing Marr until we have confirmed and announced a date for the Neil interview.”
“We can’t take a Sunday morning programme which is already balanced in its own right as being in mitigation against not doing Neil. Andrew Neil is our priority,” another source at the broadcaster said.
Additional reporting by Press Association.