According to The Times and The Daily Telegraph, Rowan Williams argued in the letters, written in 2000 and 2001, that biblical prohibitions on homosexuality were targeted at "heterosexuals looking for sexual variety in their experience."
"I concluded that an active sexual relationship between two people of the same sex might therefore reflect the love of God in a way comparable to marriage, if and only if it had about it the same character of absolute covenanted faithfulness," Williams was reported to have written.
The letters were apparently written to Deborah Pitt, a psychiatrist and evangelical Christian, while Williams was still archbishop of Wales.
He was appointed Archbishop of Canterbury, the head of the Church of England and the leader of the world's 77-million-strong Anglican Communion, in 2002.
Williams presided over the once-a-decade Lambeth Conference earlier this month, amid a row over gay bishops in the church.
Around 200 bishops boycotted the meeting over the consecration five years ago of the first openly gay bishop, Gene Robinson, the bishop of New Hampshire in the United States.
According to the newspapers, Williams apparently wrote that the church "has shifted its stance on several matters -- notably the rightness of lending money at interest and the moral admissibility of contraception so I am bound to ask if this is another such issue."
In response to the reports, Lambeth Palace, the Archbishop of Canterbury's office, only quoted remarks given by Williams recently to the Church of England Newspaper: "When I teach as a bishop I teach what the church teaches. In controverted areas it is my responsibility to teach what the church has said and why."