After suffering a serious knee injury which has taken over 18 months to recover, Odubajo could be looking to put his career back on track with a move to Celtic.
Celtic boss Brendan Rodgers has predicted that the right-back could be a steal for the club, if he can prove his fitness.
Odubajo's contract with Hull City expires this weekend, and the 24-year-old has travelled with Celtic players to their pre-season camp in Austria to discuss the possible move.
Rodgers has stressed that the former England cap would need to be fully fit and not showing any ill-effects from the injury, if the club were to take him on.
If all goes well though, the Celtic boss thinks the deal could be a good bit of business for the club.
"It is a boy we know and we have had a conversation with," Rodgers said.
"It is nothing more than that. We are looking to develop our game and the way we work and his profile, when he was fully fit, was really exciting.
"He is coming back from a really awful injury. We are just taking the opportunity to have a chat to see where he is at and now we will take it from there.
"Moses if he is a fully fit player and up to speed then he is a big, big talent. I saw him at Brentford and Hull and he was an exceptional talent.
"He was a winger that has been turned into a full back so his intent to get forward is there but he is also good defensively as well. That sort of player attracts us.
"We are looking to develop he squad. He will be leaving Hull. We got permission to speak but obviously there is a concern over the fitness. There is still a lot to go before that could happen but it was just nice to talk him and see where he was at."
Drawing a parallel, Celtic took on Craig Gordon four years ago despite his injury history, Rodgers went on: "Absolutely, as a coach you have to look.
"I have always been one to give opportunities whether it has been through injury or they have been discarded."