The investigation stems from the brief existence of a website that was created to sell Fournette merchandise. The site was launched the week of last year's season opener and was taken down within 24 hours, according to USA Today.
According to the report, the site was selling merchandise featuring Fournette's catchphrase "BUGA Nation." BUGA stands for "Being United Generates Attitude," a slogan invented by the Fournette family.
USA Today reported that Paul Price, described as the Fournette family's manager, made $10,000 worth of payments to build a website and make the site's merchandise.
It is against NCAA rules to use a player's image or likeness, and the website skirted that issue by using only the catchphrase.
LSU coach Les Miles skirted the issue when asked about it Wednesday.
"I know very little if anything noteworthy," Miles said. "What I do know is positive. That's all I can say."
Fournette is considered to be a Heisman Trophy finalist this season behind a nation-leading 1,383 rushing yards, and he has already become big business. A game-worn Fournette jersey sold for $101,000 at auction; the funds are earmarked for helping victims of the flood in South Carolina.
The ninth-ranked Tigers host Arkansas on Saturday.
Information from ESPN.com's David Ching contributed to this report.