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David Cameron has full confidence in Andrew Mitchell and said he had taken the "correct action" in apologising for his outburst at police, says No 10.

The chief whip told reporters he was sorry for not showing enough respect to the police, but maintained that he "did not use the words attributed to me".

He said he wanted to "draw a line" under last Wednesday's incident.

The Police Federation and Labour both want an inquiry, while some Lib Dems were unimpressed by the statement.

Mr Mitchell is reported to have sworn at an officer during the incident on Wednesday evening - when police directed him away from the main gate to a smaller pedestrian gate - and said "learn your place" and "you don't run this government".

'Discourteous'

In his first statement to reporters about the incident, on Monday morning, Mr Mitchell said that the incident had come at the end of a "long and extremely frustrating day".

He added: "I didn't show them the amount of respect I should have done."

He said the police officer involved had accepted his apology.

When asked whether he had used the word "plebs", as is alleged in newspaper, Mr Mitchell did not answer directly, saying only that he had not used the words that have been reported.

The newspaper says it has seen the official police report of the incident, which does include the word.

Chair of the Metropolitan Police Federation John Tully told the ����ý that Mr Mitchell's apology was not sufficient and an inquiry was needed to resolve the issue.

"He is calling the officer's integrity into question and that isn't acceptable from my point of view.

"I think the Downing Street officials should launch an inquiry as soon as possible to get to the truth."

The issue dominated the prime minister's official spokesman's lobby briefing on Monday, especially the apparently conflicting accounts given by Mr Mitchell and by the police.

Asked about the differences, the spokesman said the PM "obviously accepts the statement" from Mr Mitchell.

It has also emerged that the Metropolitan Police is investigating how police records of the incident were obtained by the newspaper.

The force confirmed they had not made a complaint to Downing Street and they were satisfied with how the officers involved dealt with the incident.

Deputy Prime Minister Nick Clegg - who had called for Mr Mitchell to give a full account of the row - said Mr Mitchell had been right to apologise and show contrition.

He said: "Being discourteous is a bad thing. He said the police officer has accepted his apology, which is significant. It is discourteous and just wrong to be rude to police officers."

Former Lib Dem leader Sir Menzies Campbell said: "If Andrew Mitchell's determination was to draw two lines under this matter he hasn't achieved it."

And fellow Lib Dem Jeremy Browne, a Home Office minister, said: "I think people want to know what was said... explaining to the media what was not said, is not the same as explaining to the media what was said."

Business Secretary Vince Cable worked in a joke about the row during his speech to the Lib Dem conference, referring to David Cameron and Boris Johnson's school days at Eton, before adding: "I've been told that jokes about social class are not good for the unity of the coalition. But as a mere pleb, I couldn't resist it."

Labour's shadow policing minister David Hanson said: "If Number 10 don't believe the testimony of a number of Downing Street police officers that's very serious.

"Alternatively if the chief whip isn't telling the prime minister the full story that's equally as serious.

"The prime minister must ensure there is a proper investigation by the cabinet secretary and Andrew Mitchell needs to deliver a lot more answers very fast."

Labour MP Bill Esterson has written to the Parliamentary Commissioner for Standards, Sir John Lyon, asking him to investigate whether Mr Mitchell has broken the code of conduct for MPs.