Dr John Sentamu told them he was "deeply ashamed" at the Church of England's failure to tackle abuse and protect young children.
The men were abused by the Very Revd Robert Waddington, the former dean of Manchester Cathedral, over a 60 year period.
Dr Sentamu will soon publish a report into the abuse which is expected to outline failures of the Church to pick up on the abuse.
In the letter he said he was "determined that we in the Church of England should face up to the wrong which has been allowed to be done to those children who were abused by Robert Waddington."
Waddington was stripped of his right to conduct services by former Archbishop of York, Lord Hope of Thornes, but he was not reported to the police or social services.
The Archbishop added: "Apologies given years later are unlikely to be satisfactory, but I want you to know that for my part I am deeply ashamed for those times when the church has failed to either listen or to act where children were at serious."