The 17-year-old is highly rated at the Premier League club and featured for the first team in pre-season friendlies against Real Madrid, Bayern Munich, Inter Milan and Juventus.
The Italian champions were impressed by the Republic of Ireland youth international and will continue to track his development, Calciomercato reports.
Parrott impressed on his debut for the Irish Under-21s during the recent international break, scoring three goals in two games against Armenia and Sweden.
He has been prolific at youth level for Spurs too, notching 22 strikes in 31 matches between U18, U19 and U23 levels.
The striker is yet to make his first-team debut for the north Londoners, but boss Mauricio Pochettino has urged patience.
"He's going to be involved sometimes - perfect. He's going to be involved with the U23 - perfect. Sometimes he's going to train with us - good," the Tottenham manager said.
"I think now is a process that he needs to be relaxed. If we put too much pressure [on him] we're going to push him to make a mistake and that is what we don't want."
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