Robert Hannigan stood down as director of the spy agency in 2017 after less than three years in the post, citing "family reasons".
The Mail on Sunday reports his departure followed the discovery that in 2013 he had given a character reference on behalf of a Catholic priest charged with possessing child pornography.
The priest, who was said to have been a long-standing family friend of Mr Hannigan, was given a non-custodial sentence and went on to offend again, the paper said.
Mr Hannigan's involvement in the case was said to have been discovered during a major investigation into online chatrooms by the National Crime Agency.
Mr Hannigan told the Mail the priest had been a "close family friend" for 20 years and they had submitted a character reference to the court "in good faith" after he pleaded guilty to the offences.
"His subsequent criminal actions appalled us and have shown that our judgment was completely wrong," he added.
"When I later became director of GCHQ, all the correct steps were taken in relation to my involvement in this case and this was verified by Government lawyers.
"This is a personal family matter. We will not be making further comment."
However, SNP MP Pete Wishart said the case "reeked of a cover-up" and called on Theresa May to explain what ministers knew.
"No 10 must give a full account of what they knew, when they knew and why they have failed to do so before," he told the Mail.
"The Prime Minister must come to the Commons and explain the Government's role in this appalling episode."
The Cabinet Office declined to comment on the report.
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