Kean posted on Instagram images of some of the abuse he received from Inter fans in the aftermath of Monday's 2-1 loss at the San Siro, with the message "Still happening, in 2025..."
Fiorentina said in a statement the online offenders "have been reported to the relevant authorities."
"The Viola stand with Moise Kean who was the victim of serious racial abuse after the match at Inter in Milan," the club added.
Kean has scored 15 times in Serie A since moving to Fiorentina from Juventus last summer, a 13 million euro signing which was seen as a gamble but that has paid off for both player and club.
Two of those strikes came in Fiorentina's 3-0 thumping of Inter in Florence on Thursday when the two teams completed a fixture which had been stopped in December due to Edoardo Bove's on-pitch heart attack.
Kean has played himself back into the Italy reckoning after not scoring at all for Juve last season, with 19 goals in all competitions for Raffaele Palladino's exciting team.
That tally is already the highest in a single season of his career and he sits second in the Serie A scoring charts behind Azzurri teammate Mateo Retegui who plays for Atalanta.
Kean has been racially abused by fans of Serie A clubs before, first in 2019 at Cagliari whose hardcore supporters have been repeat offenders in a country where far-right fan groups are common and racial discrimination from the stands is a semi-regular occurrence.
Fiorentina were also punished with a suspended one-match stand closure in November 2023 after their fans targeted Kean, Dusan Vlahovic and Weston McKennie while playing in Florence for Juventus.