Gibbs, 26, was another member of the victorious 2005 side, and has spent his entire NRL career with the club, racking up well more than 100 appearances.
Although Gibbs is under contract until the end of 2014, Fulton revealed to Australian magazine Rugby League Week his team-mate could follow him to another club in a package deal.
He said: “We’ve got the same manager and it would be ideal that we stay together.
“Whether we stay at the Tigers or (go) somewhere else – we’ll leave that to the coaching staff and everyone else.
“(Wests Tigers) is the club I love. All my mates are here, all my family are Tigers supporters now and to leave – I’d be upset.”