Police stood by reports that Mitchell – furious at being told to get off his bicycle and use a pedestrian gate rather than Downing Street's main gate – had called officers "morons" and "plebs".
As David Cameron condemned the outburst, Mitchell apologised in a telephone call to the officer, believed to be a member of Scotland Yard's diplomatic protection group, SO6. The MP for Sutton Coldfield said in a statement that he did not use any of the words reported but accepted: "I did not treat the police with respect."
Mitchell, 56, was reported in the Sun to have told officers: "Best you learn your f***ing place. You don't run this f***ing government. You're f***ing plebs." John Tully, chairman of the Metropolitan Police Federation, said the remarks were "outrageous". He had made inquiries and he believed the newspaper's report.
"I know what the officers have told me, and what was reported in the Sun this morning is absolutely what happened," he told Sky News. Although he had not spoken directly to the officer concerned, he had spoken to federation representatives.
He added: "He [Mitchell] should resign. As a cabinet minister, it's unacceptable for someone of his standing to use such disrespectful and abusive language to a police constable, let alone anyone else. If the shoe was on the other foot and my officer had said those things he'd be out of a job now. It's double standards."
Labour said that No 10 must say whether Mitchell was accusing the police officer of lying. "Downing Street must make clear exactly what Andrew Mitchell said to the police officer," a spokesman said.
"There are two alternatives: either the chief whip used appalling and offensive language to an officer going about their duty, or Mr Mitchell is saying the officer is lying.
"Downing Street will know. They must make the position clear urgently. A half-hearted apology is not enough."
Labour called for police incident records to be revealed to show whether Mitchell, a former shadow police minister, was telling the truth.
Speaking on a visit to Manchester, in the wake of the murders of PCs Fiona Bone and Nicola Hughes, Cameron said: "What Andrew Mitchell said and what he did was not appropriate. It was wrong. It is right that he has apologised."
Paul McKeever, chairman of the Police Federation, which represents rank-and-file officers in England and Wales, said it was "hard to fathom how someone who holds the police in such contempt could be allowed to hold a public office".
He added: "Mr Mitchell's half-hearted apology for the comments made whilst leaving Downing Street will do little to build bridges with the police, who feel they have once again been treated with a lack of respect and civility by members of this government.
"The lack of regard that some within government appear to hold police officers in is especially disappointing during this tragic week for the service and does nothing for [their] the rock-bottom morale of officers in this country."
The shadow home secretary, Yvette Cooper, described the reported outburst as an "utter disgrace".
"Downing Street clearly have a lot of questions to answer. They must make clear exactly what happened, including what was said and whether the deeply offensive language reported was used," she said. "How can a chief whip hope to do his job or instil respect when he behaves like this and can't even keep his cool?"
The former Conservative party chairman Lord Tebbit denounced Mitchell's comments as "extraordinarily stupid" and urged Cameron to have a "heart-to-heart talk" with his new chief whip.
The London mayor, Boris Johnson, told last year's Conservative party conference: "If people swear at the police, they must expect to be arrested."
Joanne McCartney, Labour chair of the London assembly's police and crime committee, said: "It seems it's one rule for the public and another for those running the country. Perhaps the prime minister should do more than accept a simple apology."
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