A lawyer for three victims of Peter Ball, the disgraced former Bishop of Lewes and Bishop of Gloucester, made the claim to Dame Lowell Goddard's public inquiry into child sex abuse.
Mr Ball, 84, was jailed in October 2015 for 32 months after pleading guilty to a string of historical sex offences.
He had previously been investigated in 1993 and given a caution for gross indecency by police after abusing a trainee monk.
Lawyer, Richard Scorer, who represents the three victims, made the claim about Lord Carey while an application for the men to be "core participants" in the inquiry at a hearing in the Royal Courts of Justice was made.
He said one man, known as A13, had complained in 1992 to the then Archbishop, the most senior figure in the Anglican Church.
Mr Scorer said: "A13 can tell the inquiry about a very detailed complaint he made to Archbishop George Carey in 1992, reporting Peter Ball's behaviour ... years previously.
"We believe that the Archbishop of Canterbury failed to pass that information on to the police and is one reason, we believe, a proper investigation of Peter Ball's behaviour and abuse was delayed by over 20 years."
One section of Dame Lowell's inquiry will investigate child sex abuse by members of the Anglican Church and ecclesiastical authorities' reaction and response to it.
The investigation is looking in detail at the allegations that involve the Diocese of Chichester - where Peter Ball worked as Bishop of Lewes from 1977 to 1992, when he took the position in Gloucester.
Lord Carey has previously denied he covered up Peter Ball's crimes.
Following the jailing of Peter Ball last year, Lord Carey said outside the Old Bailey: "I greatly regret the fact that, during my tenure as Archbishop of Canterbury, we dealt inadequately with Peter Ball's victims and gave too much credence to his protestations.
"Allegations by some that my actions amounted to a cover-up or collusion with the abuser are wrong. I have always insisted upon the highest standards of holiness of life from all who are ordained."
During his time as the Bishop of Lewes, Peter Ball chose 18 vulnerable victims to commit acts of "debasement" in the name of religion, such as praying naked at the altar and encouraging them to submit to beatings, his trial heard.
Complaints were made but Mr Ball went on to become Bishop of Gloucester and was never charged.
He continued to work as a priest in Truro after he accepted the caution.
Following Wednesday's preliminary hearing, chaired by Justice Goddard, the Church of England has issued the following statement: "We welcome the plans outlined in today's preliminary hearing by Justice Goddard, for the Anglican Church, as it examines the extent to which institutions and organisations in England and Wales have taken seriously their responsibility to protect children.
"As a church we will be offering full cooperation and are committed to working in an open and transparent way, with a survivor-informed response. We are already reviewing our 2008 Past Cases Review, referred to in today's hearing.
"We commend those survivors who have come forward as core participants or who are contributing to the Truth Project, knowing how costly it will be to relive their experiences. It goes without saying we will examine lessons learnt from the Inquiry's findings and believe it will play a vital part in our commitment to making the Church a safer place for all.
"The true cost of child abuse and the abuse of adults at risk is far higher than any of us have ever been prepared to acknowledge in terms of the mental, emotional, social and physical health and well-being of very large numbers of our population."
