Pressure is mounting on Conservative minister Andrew Mitchell after police leaders urged him to resign and Labour called for a full account of his outburst at an officer.
The Tory Chief Whip denies claims he swore at a policeman on duty outside Downing Street and called him a "pleb".
The officer concerned has insisted reports of what happened are accurate.
Mr Mitchell has apologised but Labour said No 10 "must make clear" the exact words he used in the confrontation.
The body representing rank-and-file police officers said Mr Mitchell's alleged remarks were "outrageous" while the prime minister said the minister's conduct was "not appropriate".
John Tully, the Metropolitan Police Federation chairman, said the minister's outburst was "disgraceful" and he must resign, adding that the minister was "lucky not to be placed under arrest if indeed he did say those words, and I have no reason to doubt that he did".
Mr Mitchell's behaviour has also been criticised by a number of Conservative colleagues.
'Lack of respect'
The incident occurred on Wednesday after Mr Mitchell, MP for Sutton Coldfield, was told by officers to get off his bicycle as he left Downing Street and use the smaller pedestrian gate instead of the main entrance used by cars.
He is reported by the Sun to have used foul language and told the officer at the gates to "learn your place" and "you don't run this government".
The officer - a member of Scotland Yard's Diplomatic Protection Group, SO6 - reported the incident to his superiors.
The BBC's home affairs correspondent Danny Shaw said the officer had backed up the Sun's version of events and the language - including the word "pleb" - that Mr Mitchell was reported to have used.
Downing Street said Mr Mitchell had "apologised profusely" to the officer on the telephone and sources also stressed Mr Cameron had made clear to Mr Mitchell how displeased he is at the confrontation with a police officer.
The source added that the PM still has faith in Mr Mitchell as Chief Whip and that the minister disputes The Sun's version of events.
Mr Cameron said: "He has obviously apologised to me, but more importantly he has apologised thoroughly to the police and that needed to be done."
The prime minister also praised the police, saying they do an "outstanding job".
But Mr Mitchell's apology has not been accepted by police union leaders.
Police Federation national chairman, Paul McKeever, said: "It is hard to fathom how someone who holds the police in such contempt could be allowed to hold a public office.
"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 the rock bottom morale of officers in this country."
Backbench Conservative MP Philip Davies said the reported comments were "obviously unacceptable" and that he would tell Mr Mitchell, who attends cabinet but is not a full cabinet minister, so to his face when he next sees him.
And a senior backbencher, who spoke to the BBC on condition of anonymity, said the comments were "not out of character" and Mr Mitchell should consider his position.
"I am deeply shocked that a senior member of the government could speak to an officer in this way," he said.
Asked if Mitchell's position was tenable, he said, "I think it's very difficult for him to continue".
Labour have said the reported comments were "appalling" and No 10 has "a lot of questions to answer".
"Downing Street must make clear exactly what Andrew Mitchell said to the police officer," a party 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."
Mr Mitchell only became Conservative chief whip earlier this month, after being moved from his previous position as international development secretary in Mr Cameron's first major reshuffle.
In his current role, he is responsible for enforcing party discipline and keeping rebellious backbenchers in line.