Ian Sawyer from Manchester said: "I'm homosexual, I'm not a paedophile. Does Mr Fury know the difference?"
Fury has been quoted as saying a woman's "best place is on her back" and criticised homosexuality and abortion.
The boxer, also from Manchester, has denied making any homophobic comments.
He said: "Tyson Fury loves his fellow humans. He doesn't hate anybody."
Mr Sawyer told the Victoria Derbyshire programme he was offended "to such an extent that I went to Greater Manchester Police" but has yet to be interviewed by police for an official statement.
He said 27-year-old Fury, who became world champion by beating Wladimir Klitschko last month, was entitled to his opinions, but "if they are offensive he should keep them private".
"After he won the championship I woke up thinking how wonderful it was to have a world champion in Manchester at a sport that I've admired since Muhammad Ali and then when I woke up yesterday morning and listened to his words [on the Victoria Derbyshire programme] I thought he had gone from hero to zero," he added.
In an interview in the build-up to his world title fight, Fury, who comes from Wythenshawe, said: "There are only three things that need to be accomplished before the devil comes home: one of them is homosexuality being legal in countries, one of them is abortion and the other one's paedophilia.
"Who would have thought in the '50s and '60s that those first two would be legalised?"
Fury has been stripped of his IBF belt because he will not be fighting the organisation's mandatory challenger Vyacheslav Glazkov in his next fight, instead planning a rematch with Klitschko.
The Ukrainian former champion has exercised a rematch clause in the contract for his original fight with Fury, who still holds the WBA and WBO belts.
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A man who made a hate crime complaint to police about world heavyweight boxing champion Tyson Fury has said Fury is "ignorant to the difference between homosexuality and paedophilia".