Cousin has only been with Rangers for just over a year after spells with Martigues, Niort, Le Mans and Lens in France. He joined the Scottish Premier League side for £750,000.
The 31-year-old scored 11 league goals in 21 starts and 7 substitute appearances for the Gers. The last of those appearances was the 4-2 derby win against Celtic yesterday, a game he scored in before being sent off for two bookable offences. That means that, if he has signed for Hull City, he will be suspended for the game at Newcastle before being able to make his debut against Everton at the KC Stadium.
City have already signed defenders Paul McShane and Kamil Zayatte on loan ahead of the transfer window closing.