The 33-year-old defender has been on the books of French club Marseille since 2011 and is out of contract at the end of the month.
According to a report in The Guardian in April, Liverpool are interested in the services of right-back Fanni.
The same British newspaper claimed earlier this week that the Reds’ Premier League rivals Crystal Palace want to sign the defender on a free transfer this summer.
Marseille chief Labrune has now stated that the Ligue 1 outfit have made an offer to Fanni to extend his stay at Stade Velodrome.
“We want to keep him (Fanni),” Labrune told RMC, as translated by Get French Football News. “We have made an offer. It is up to the player.”
Fanni is an important player for Marseille and made 27 appearances in Ligue 1 in the recently concluded 2014-15 season.
The France international scored two goals and created seven chances in those games, helping L’OM to a fourth-placed finish in the league.
The former Lens star had a pass accuracy of 86% and a shot accuracy of 60%, and won 58 headers, 16 dribbles and 50 tackles.