The Government figures held a number of phone calls and communications with Buckingham Palace about the prospect, the BBC report says.
The claims are featured in the second episode of the Laura Kuenssberg: State Of Chaos series, exploring the turmoil in Westminster between 2016 and 2022.
Tackling coronavirus in May 2020, significant tensions were spilling over between Mr Johnson’s political team and the Civil Service.
Sources told the documentary that senior officials expressed their fears about the then prime minister’s conduct to the palace.
The officials reportedly hoped the then Queen would raise her concerns with Mr Johnson during their private audiences.
One source claimed Mr Johnson “had to be reminded of the constitution”.
Dominic Cummings, who served as his top adviser, and the then Civil Service chief Sir, now Lord, Mark Sedwill, clashed more than once.
Helen MacNamara, a former deputy cabinet secretary, declined to discuss the calls but said they had been “systematically in real trouble” during the period.
After Mr Johnson was treated in hospital for Covid-19, she said there was “extreme” talk in his political team about the failings of Whitehall.
She said they were taking a “kind of smash everything up, shut it all down, start again” attitude.
Mr Johnson had already caused trouble with the palace when in 2019 he suspended the Commons for five weeks ahead of the Brexit deadline.
His advice to “prorogue” Parliament to the then Queen was found to have been unlawful by the Supreme Court.
Buckingham Palace and Downing Street have been asked to comment.