"These cases must be brought to our attention so that investigations can be carried out," Michel Delpuech said on Europe 1 radio. "Our services will obviously pursue any aggressors without fail if they are identified."
Reports of unsolicited kisses and groping began appearing on social media after fans poured onto French city streets to celebrate France's victory over Croatia in Sunday's final.
And there were more during the team's victory parade down the Champs Elysées on Monday, spawning the hashtag #MeTooFoot, a reference to the MeToo movement sparked by allegations against Hollywood moghul Harvey Weinstein.
"For me it was during the third goal... when I felt a hand slide up under my skirt," a Twitter user named Isa posted Monday.
Other victims said they had not filed charges, saying they did not believe the police would pursue their cases since their assailants vanished into the crowd.
Equality Minister Marlene Schiappa also urged victims to come forward on Tuesday, saying on Twitter that "A man forcibly kisses a woman during the World Cup parties, that's sexual assault which is punished by the law."