An Italian newspaper published extracts at the weekend of an interview Tavecchio had given to sports website SoccerLife.it last June, shortly before he became head of the FIGC.
In it, he said it was better to keep Jews "at arm's length", then went on to say that while he had "nothing against" homosexuals, he wanted them kept "far away from me".
Tavecchio said the interview was taped without his knowledge and questioned whether the recording had been manipulated. He also accused the website of trying to blackmail him, something SoccerLife has denied.
Flavio Romani, head of Italy's Aricgay movement said Tavecchio should be kept "far away from a position of responsibility in sport", while the head of Rome's Jewish community called on Italian sport "to take a step back".
Tavecchio was barred last year from holding any position with world governing body FIFA for six months after he had made a comment about a fictitious African player he named Opti Poba "eating bananas".
The comment, made before he took charge of the FIGC, came as he was addressing the question of the lack of opportunities for young Italian players at professional clubs.
The latest controversy has prompted more outrage.
"The words which have emerged through the audio recording are unacceptable and must be condemned," said Giovanni Malago, head of the Italian Olympic Committee (CONI).
Malago said that there was no mechanism for CONI to take action against Tavecchio although he had asked the FIGC to investigate the case, including the allegations made by Tavecchio.
"I believe that each person must answer to their conscience and adopt the most appropriate behaviour," added Malago.
(Writing by Crispian Balmer; additional reporting by Brian Homewood in Zurich; Editing by John O'Brien)
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