Neil said that if the Prime Minister is expected to face the likes of Donald Trump and Vladimir Putin then he should be able to sit down for a half-hour interview with him.
Mr Johnson has been accused of "running scared" from scrutiny by avoiding in-depth questioning from the veteran broadcaster, even though rival leaders, including Mr Corbyn, have taken part.
• READ MORE: General election: Why Boris Johnson will be worse than Margaret Thatcher – Tommy SheppardThe Prime Minister, who has also refused to do an interview on ITV, will debate Mr Corbyn on the BBC on Friday night, with less than a week to polling day.
Some 6.7 million people tuned in when the pair took part in a head-to-head debate on ITV last month.
On Thursday night, Neil challenged Mr Johnson to finally agree to an interview on the theme of "trust" and why voters have "deemed him to be untrustworthy".
Addressing viewers directly following his interview with Brexit Party leader Nigel Farage, Neil said he wanted to put "questions of trust" to the prime minister.
• READ MORE: Opposition is missing open goal at heart of Boris Johnson’s Brexit-dominated campaign – Joyce McMillan"Questions we'd like to put to Mr Johnson so you can hear his replies.
"But we can't, because he won't sit down with us.
"There is no law, no Supreme Court ruling, that can force Mr Johnson to participate in a BBC leaders' interview.
"But the Prime Minister of our nation will, at times, have to stand up to President Trump, President Putin, President Xi of China.
"So it was surely not expecting too much that he spend half an hour standing up to me," he said.
Neil, who outlined policy issues on which he would question Mr Johnson, said leaders' interviews had been a key part of the BBC's prime-time election coverage for decades, adding: "We do them on your behalf to scrutinise and hold to account those who would govern us. That is democracy.
"We've always proceeded in good faith that the leaders would participate. And in every election they have. All of them. Until this one.
"It is not too late. We have the interview prepared. Oven-ready, as Mr Johnson likes to say.
"The theme running through our questions is trust, and why at so many times in his career, in politics and journalism, critics and sometimes even those close to him have deemed him to be untrustworthy.
"It is, of course, relevant to what he is promising us all now."
The Tories declined to comment.
Liberal Democrat leader Jo Swinson said: "Boris Johnson must stop ducking scrutiny.
"His cowardly behaviour shows why he simply isn't fit to be prime minister."
Earlier, ITV confirmed that Mr Johnson has refused to participate in their series of leader interviews with Julie Etchingham.
ITV said they contacted Mr Johnson's press team on "repeated occasions" with times and dates, but that his team confirmed on Thursday that he will not be taking part.
Ian Lavery, Labour Party chairman, said: "Boris Johnson thinks he's born to rule and doesn't have to face scrutiny.
"He's running scared because every time he is confronted with the impact of nine years of austerity, the cost of living crisis and his plans to sell out our NHS, the more he is exposed."
Meanwhile, the Labour Party has complained to the BBC's director general, accusing the broadcaster of "slanted and biased" election coverage.
In a letter, Labour's co-campaign coordinator Andrew Gwynne raised concern about Mr Johnson's "failure" to be interviewed by Neil.
Mr Gwynne said the party agreed to Mr Corbyn's interview with Neil based on the "clear understanding" that Mr Johnson had agreed the same terms.
Editor of Channel 4 News, Ben de Pear, said his programme was also awaiting confirmation from Mr Johnson regarding an interview.
Elsewhere, shadow chancellor John McDonnell said Labour would not negotiate with smaller parties in the event of a hung parliament.
"We will implement our manifesto...no negotiation, no deal, no coalitions," he told the Financial Times, adding: "We'll roll out our programme and let's see if the Lib Dems vote against the real living wage at £10 an hour, let's see if the SNP vote against the proposals we brought forward for ending austerity."