The Swiss centre-back is currently on a season-long loan spell with Bundesliga outfit Hamburg and it is unlikely he'll return to first-team action for the Gunners next season.
Tigers boss Steve Bruce knows the international defender well, having managed the player as a youngster when Birmingham loaned him from his parent club.
With just 86 appearances in an Arsenal shirt in 10 years at the club, Djourou is expected to make a permanent move away from the Emirates this summer.
And should Hull beat the drop, Bruce will line up a £2.6million bid.
Djourou moved to North London on scholarship forms in 2004, but has failed to have a major impact.
His Emirates departure would follow the same path as international colleague Philippe Senderos who also joined Arsenal as a youngster before failing to make the grade.
The Gunners defender still has 15 months on his current deal, although with all parties keen on a deal, it appears likely that the Switzerland star will move.