The England international is currently in the middle of a season-long switch with the French champions from Liverpool.
Cole has been a regular in the Lille starting lineup this season and Mavuba is keen for the 30-year-old to extend his stay in Ligue 1.
"I think Joe really wants to stay because he takes pleasure playing for this team," Mavuba told RTL.
"Now it is up to the board to work it out with him."
Cole has made 56 appearances for England, scoring ten goals.