Lib Dem minister Chris Huhne has said allegations he asked someone else to take his penalty points for a 2003 speeding offence are "simply incorrect".
Downing Street said the prime minister had "full confidence" in Mr Huhne after it emerged police were investigating whether an offence took place.
The findings will decide whether a formal inquiry is launched.
The energy secretary said he welcomed the police investigation.
He said the allegations had been "made before and been shown to be untrue".
"I very much welcome the referral to the police as it will draw a line under the matter," he told BBC News.
"I don't want to say any more than that, I think the police can get to the bottom of this."
'In the past'
A senior Lib Dem source told the BBC that Deputy Prime Minister Nick Clegg had met Mr Huhne on Monday to discuss the weekend's allegations, which Mr Huhne denied, and that he retained the Lib Dem leader's full confidence.
Asked if Mr Huhne could stay in the Cabinet if the police do formally investigate him, a Downing Street spokesman said: "It has happened in the past."
Mr Cameron was asked if he had confidence in Mr Huhne at a press conference on the government's planned NHS shake-up. He said Mr Huhne had denied the allegations.
Liberal Democrat party president Tim Farron told the BBC's Daily Politics show that he did not think Mr Huhne would have to stand down from the cabinet if there was a full police investigation into the allegations.
He said: "If there's an investigation, there's an investigation. But he denies it. Tony Blair remained prime minister while he was investigated, I imagine Chris Huhne can just about cope with being energy secretary."
'Action where necessary'
Det Supt Wills was charged with looking into the allegations after Labour MP Simon Danczuk lodged a formal complaint with police against Mr Huhne after claims surfaced in newspaper stories following an interview with his estranged wife Vicky Pryce.
Essex Police said: "We are aware of allegations around a speeding offence which is believed to have occurred in 2003. A senior detective from the Kent and Essex serious crime directorate has been appointed to establish if this offence took place and the allegations around it.
"We take allegations such as this one extremely seriously and will take action where necessary."
Mr Huhne, MP for Eastleigh, is Energy and Climate Change Secretary in the coalition government - and was narrowly beaten to the Lib Dem leadership by Nick Clegg in 2007.
He has been in the spotlight recently for his criticism of the Conservatives over their behaviour during the recent campaign over changing the UK voting system - which saw the two government parties on opposing sides of the argument.
He confirmed last year he was splitting from his wife of 25 years, economist Vicky Pryce, and was in a relationship with another woman.
His office has denied the allegation about a speeding offence dating back to 2003, when he was an MEP, saying it had been made before and been shown to be untrue.