The prime minister has been accused of “running scared” of both Neil and a Thursday climate change debate on Channel 4, where he will be replaced by an ice sculpture.
Labour only agreed to the BBC interview after being told Johnson would do it next week - crucially after postal ballots for the December 12 election are sent out.
But Johnson has now refused to commit to even doing an interview with Neil at all, unlike all other main party leaders.
In an interview with the PA news agency, Johnson said he would continue to submit to “all kinds of inquisitions and interrogations” until polling day on December 12.
He said: “Here I am being interviewed by you, I don’t think I have barely stopped being interviewed by people since the election began and will continue to submit to all kinds of inquisitions and interrogations until polling day.”
Pressed on when he would do the Neil interview, Johnson replied: “I’m in discussion with all sorts of people about all sorts of interviews and look forward to doing many, many more.”
Asked again, he said: “I’m going to do many, many interviews over the course of the next few days and I’m sure that people are having all sorts of conversations right now about who I’m going to be talking to when and where and I look forward to it very much.”
Sign up now to get The Waugh Zone, our evening politics briefing, every weekday.
Scottish first minister Nicola Sturgeon and Jeremy Corbyn were interviewed by Neil on Monday and Tuesday respectively this week, with the Labour leader suffering a rough ride over his party’s record on anti-Semitism and its tax and spending plans.
It is understood that the BBC had planned a third special interview with Neil on Wednesday night, which was mysteriously cancelled and replaced with his regular show featuring multiple guests.
Liberal Democrat leader Jo Swinson and the Brexit Party’s Nigel Farage will face Neil next week.
Meanwhile, Johnson dismissed questions about past comments in which he branded the children of single mothers “ill-raised, ignorant, aggressive and illegitimate”.
The PM said: ““I think this is sort of breaking new records really to disinter and distort pieces that are almost 25 years old - this is a quarter of a century old, these quotes that are being culled.
“Never mind that, the reason I think it is happening is because Labour has been exposed as a party that on the central issue before the country today has no policy and it is worse than that.”