Sir Bob Neill, chairman of the Commons Justice Committee, said that if the inquiry was non-statutory, witnesses could mislead it without risking being charged with perjury or misconduct in public office.
She emphasised that such a move would show the Government was “absolutely committed to women’s safety”.
Home Secretary Priti Patel announced the inquiry earlier this week.
It is being set up on a non-statutory basis so that it can get under way swiftly rather than requiring legislation to be passed.
Even more significantly, if it was a statutory inquiry, Couzens, 48, who has been jailed for life for the abduction, rape and murder of Ms Everard, 33, could become a “core participant”, being represented at hearings, able to give statements and seeing the evidence.
However, Sir Bob told the Standard: “Although I can see the arguments for speed in a non-statutory inquiry, the ability to compel reluctant witnesses, particularly if they are perhaps serving or former colleagues of Couzens who might otherwise wish to cover things up, that would be a strong argument for a statutory inquiry to make sure those people are put on the spot and find out what they did or did not know.”
He was also nicknamed “The Rapist” by former colleagues at the Civil Nuclear Constabulary in Kent and reportedly drove around in a car naked in 2015, three years before he was hired by the Met Police.
Another former Tory minister said: “As a number of witnesses who should turn up will be police officers and former police officers, it makes sense to have a compulsion for them to give evidence. To not do this provides a loophole which could mean the full truth is not obtained.”
The Government is ready to make the inquiry statutory if officers from the Met or other forces refuse to take part.
Ministers, though, expect Met officers to fully co-operate after the force was accused of being institutionally corrupt in the way it concealed or denied its failings in investigating the murder of private investigator Daniel Morgan in 1987. A Whitehall source said: “If we need to make it statutory, we will.”
Former Home Office minister Ms Nokes also stressed that a statutory inquiry would send out a message as to how seriously the Government was taking the Couzens case.
She said: “Obviously the power to compel witnesses would be of potential benefit. This tragic killing has exposed weaknesses and misogynistic attitudes that have to be rooted out, and I want to see real determination.”
Yvette Cooper, chairwoman of the Commons Home Affairs Committee, said there are “real concerns that given the nature of this inquiry it will face delays and obstacles if it does not have statutory powers to get to the truth”.