Baroness Casey, who is conducting a review of the Metropolitan Police’s standards and internal culture, called for the inquiry in a letter to the Home Secretary.
She suggested the work could be carried out by Dame Elish Angiolini, who is already conducting an inquiry linked to the Met.
Carrick, 48, pleaded guilty to 49 offences relating to 12 women between 2003 and 2020, including 24 counts of rape.
He joined the Met in 2001 before becoming an armed officer with the Parliamentary and Diplomatic Protection Command in 2009.
In her letter to the Home Secretary she said: “I am asking you to consider that the scope of Lady Elish Angiolini’s current non-statutory inquiry should include the conduct of David Carrick and the potential opportunities the Met, other police forces and organisations may have had to identify his pattern of behaviour prior to October 2021, to stop him being a police officer and, ultimately, stop him offending.
“If this is not possible, then I consider this issue so serious that I am willing to volunteer myself to the Metropolitan Police Service to undertake this piece of work.
“We owe it to all his victims that this work takes place.”
Shadow home secretary Yvette Cooper backed Baroness Casey’s call for an inquiry into David Carrick’s case.
“Louise Casey’s right,” she said.
“There must be a full, urgent review on how serial rapist Carrick could serve as a police officer for so long.”
Carrick is facing life in prison after subjecting at least a dozen victims to appalling physical and emotional abuse over the course of 18 years, while wielding his status as a police officer to ensure they stayed silent.
The armed officer, who help to guard politicians and VIPs for more than a decade, forced the women into humiliating and degrading sexual acts, whipped one with a belt, and kept some locked for hours inside a tiny cupboard.
His case has plunged Scotland Yard into a fresh crisis, after it emerged a string of opportunities to reveal his true character were missed.
Carrick - known to colleagues as “Bastard Dave” -was finally caught in October 2021 when a woman came forward to accuse the PC of raping her in a hotel room at the end of a Tinder date.
News of his arrest opened the floodgates, with 12 more women coming forward to reveal Carrick’s catalogue of physical and sexual abuse dating back to 2003, in the infancy of his career in the Met.
“On behalf of the Metropolitan Police, I want to apologise to the women who have suffered at the hands of David Carrick,” said Assistant Commissioner Barbara Gray.
“We should have spotted his pattern of abusive behaviour and because we didn’t, we missed opportunities to remove him from the organisation.
“We are truly sorry that being able to continue to use his role as a police officer may have prolonged the suffering of his victims.”
Colleagues said Carrick’s nickname was not linked to sexual offending, but based on his reputation for “mean” and “cruel” behaviour.
He used online dating sites like Badoo and Tinder to target women, and came across initially as “fun-loving, charming, and charismatic”, investigators said.