Odubajo was pictured training with Celtic on Tuesday and could secure a full-time move to Brendan Rodgers' team.
Odubajo has already announced his departure from Hull, who paid £3.5 million to sign him from Brentford three years ago.
And that could spell a bargain for Celtic - who have also been linked with a move for Odubajo's former Hull teammate Abel Hernandez - assuming the 25-year-old is able to prove his fitness after two seasons one the sidelines.
But what do Hull fans make of the Bhoys's latest interest? This is what they're saying on Twitter...
Very decent, when fit. Pacey as anything with bags of ability going forward, and has also been known to get back and make important last ditch tackles. Been out for a while, but if kept fit could be a star.— althurstonsport (@althurstonsport) 26 June 2018
Good luck to Moses Odubajo on his trial with Celtic. The guy deserves a break after a horrendous two years. Plus playing in a league with only 4 games a season will go easy on him #hcafc pic.twitter.com/iyXB7uMEuB— Rick (@HullCityLive) 26 June 2018
Thought better of Odubajo. Taking the easy route going to Celtic #hcafc— charlie (@CharlieT_16) 26 June 2018
We was right when we called him a snake, took us for granted and shows by going to Celtic, bottled his promising career— Connor Bibby (@connorbibby123) 26 June 2018
Moses Odubajo joins Celtic on pre-season trip to Austria.— Originallambrettaman (@origlambman) 26 June 2018
Probably flog him for £10m next summer.
If he can stay injury free then he’ll definitely be a perfect fit. Gutted he left #hcafc but delighted he’s got a chance at such a good club.— Mark Spight (@markspight) 26 June 2018
Odubajo helped Hull to the Premier League in his debut campaign at the KCOM Stadium and has caps for England's Under-21s.
And the Londoner's ability to play in multiple positions - notably on the wing - could come in handy to Celtic, who have seen Patrick Roberts return to Manchester City.
Would Celtic be a good move for the former Hull player?