"In this case, permission was given for the Top Gear car to drive down Whitehall before moving to another location," it said.
"However, what the Top Gear team did on the day was not what had been agreed during the planning process.
"At no time had the BBC producers made Westminster City Council aware that the car was going to be doing anything but drive down Whitehall.
"There was no discussion between BBC producers and Westminster City Council about wheel spins and a 'donut' and permission would not have been given to do so."
The statement added: "We have spoken to the producers today to express our disappointment and we welcome the statement from Top Gear presenter Chris Evans who has said this footage will not be shown."
The BBC said it had "worked closely" with Metropolitan Police film unit and the special events unit of Westminster Council ahead of filming, with preparations spanning four months.
A statement released by the corporation said: "The Cenotaph was at no point intended to feature in the programme and therefore will not appear in the final film."
It added the driver of the car was "briefed by production prior to filming" to not do any manoeuvres close to the monument, "an instruction to which he fully adhered".
"We would like to make it absolutely clear that the Top Gear team has the utmost respect for the Cenotaph, what it stands for, and those heroic individuals whose memory it serves so fittingly," the statement concluded.
Read more:
• Top Gear host apologises for stunt
• UK chancellor tells Top Gear to be quiet
Earlier, Chris Evans said footage for the new Top Gear series filmed near the Cenotaph in London is "disrespectful" and should not be broadcast.
Speaking outside the BBC on Monday, Evans described the images as "terrible".
"They look so disrespectful," he said, before adding that there are "mitigating circumstances".
He continued: "I saw the images this morning for the first time and I felt the same as everybody else."
In response to a question as to whether it reflected positively on Top Gear, he said: "This is not a good story, no."
He added: "We're all mortified by it, so absolutely, 100 per cent, it should not be shown."
Evans admitted he does not have the final say over the fate of the footage.
"But if it was my decision, then I would say, that particular scene shouldn't be shown and I think everyone will agree," he said.
LeBlanc was seen driving around Westminster as shooting took place for the new series of the BBC Two show, which is due to air in May, and photos show large tyre circles left on streets surrounding the war memorial after the stunt.
Speaking on his BBC Radio 2 show, Evans said he "completely understood the furore" around the photographs and admitted it had been "unwise" to film anywhere near the Cenotaph.
He said: "It doesn't matter what actually happened, it doesn't matter what the circumstances were that could explain this away. What is important about this is what these images look like, and they look entirely disrespectful which is not and would never be the intention of the Top Gear team or Matt (LeBlanc).
"On behalf of the Top Gear team and Matt, I would like to apologise unreservedly for what these images seem to portray.
"There have been some very incendiary comments written alongside these pictures and I completely understand this furore, but the Top Gear team would never ever do that.
"Retrospectively it was unwise to be anywhere near the Cenotaph with this motor car."
Evans said LeBlanc had been filming in a car called the Hoonicorn alongside racing driver Ken Block and filming could continue on Monday and Tuesday.
The stunt has been described as "gravely disrespectful" by retired Colonel Richard Kemp.
He told the Telegraph: "This is a sacred tribute to millions of people who have done far more for their country than Chris Evans and Matt LeBlanc ever will."
He added: "Jeremy Clarkson was certainly no saint but I don't believe he would have ever performed a stunt in such bad taste."
- PAA
