The Tory enforcer concedes that he said "f******" when a member of Scotland Yard's Diplomatic Protection Group, SO6, refused to let him cycle out through the main Downing Street gates.
Mr Mitchell is reported to have said: "Look, I'm the chief whip, I work at Number 9 (Downing Street)," before muttering: "You guys are supposed to f****** help us."
An unnamed friend of the minister told the Sunday Telegraph: "He does not dispute he lost it a bit. It was in frustration at the episode and not aimed directly at the officers. It was the fourth time he had been at Downing Street that day - he is frequently allowed to use the main gate on his bike. He is absolutely not accusing anyone of lying."
When allegations about the tirade were published on Friday, Mr Mitchell issued a statement denying using "any of the words that have been reported".
The friend added: "He realises there may be differing versions of what was said but he is adamant he did not use the words he is reported to have used."
Mr Mitchell has faced calls to resign over the angry rant, which proved acutely embarrassing for Prime Minister David Cameron when details emerged as he headed to Manchester on Friday to pay his respects to murdered Pcs Nicola Hughes and Fiona Bone.
Mr Cameron was told to take responsibility for the Tory enforcer's behaviour by the widow of murdered police officer Sharon Beshenivsky and Labour called on him to "urgently" review the evidence.
Paul Beshenivsky, whose wife was killed on duty in Bradford, West Yorkshire, in November 2005, told Sky News: "David Cameron should be responsible for what his ministers say, they are his ministers and they are working with him.
"Ministers shouldn't be going round foul-mouthing police officers, especially under the current circumstances."
John Tully, chairman of the Metropolitan Police Federation, has called on Mr Mitchell to quit over his "abusive" outburst.
But seasoned politician Ken Clarke, minister without portfolio, defended his colleague
He said: "I have known Andrew for a long time and he is a perfectly reasonable, courteous man with the same high regard for the police services as anyone else.
"He obviously had a flare of bad temper on this occasion and has rightly apologised. I do think this should be allowed to set the matter at rest."
Mr Mitchell, a keen cyclist, was reported by the Sun to also have called the police "morons".
"Best you learn your f****** place. You don't run this f****** Government. You're f****** plebs," it reported him as saying.
In a statement on Friday the Tory, who was also a minister under John Major in the early 1990s, said: "On Wednesday night I attempted to leave Downing Street via the main gate, something I have been allowed to do many times before.
"I was told that I was not allowed to leave that way. While I do not accept that I used any of the words that have been reported, I accept I did not treat the police with the respect they deserve.
"I have seen the supervising sergeant and apologised, and will also apologise to the police officer involved."
Deputy Liberal Democrat leader Simon Hughes said it was "never good" for a public figure to be disrespectful to the police.
He said: "I wasn't there and I don't know exactly what happened and there appears to be some dispute as to exactly what happened.
"Clearly it is never good if any person in public office is shown to be disrespectful of the police.
"The police aren't always perfect, they don't always get everything right but there's a basic principle we should follow which is that the police are there to be respected because of the amazingly important work they do for us.
"At this time, of all times, still in the seven days following the death of two police constables on duty, I think the message has to be that we respect the police and we don't lose our tempers with them."
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