Barnett (33-7) beat Hunt via submission when the pair met in the now-defunct Pride promotion in Japan in 2006.
"I love rematches," Hunt told overseas podcast The MMA Hour. "I've never lost a rematch, Josh Barnett, me and you my friend, we've got a date. He's No 5 in the world, I want to fight guys in the top 10. I'd like to fight and get close to that title shot, That's all I'm interested in; my whole purpose is to fight for the world title."
UFC Fight Night 55 will be headlined by middleweight contenders Luke Rockhold and Michael Bisping but a Hunt v Barnett co-main event would be a massive addition to the card.
It seems unlikely the fight would be booked as Barnett, who was last seen in the UFC in December when he lost via knockout to Travis Browne, has given little indication of when he would like to return to the octagon.
Most of the other top-10 heavyweights are involved in bouts during the next few months.
Hunt returned to Sydney yesterday and will spend a week across the Tasman before heading back to Auckland where he will return to training with coaches Steve Oliver and Lolo Heimuli as he awaits his next engagement.