The England Under-21 international has been heavily linked with both Newcastle United and Brighton & Hove Albion, who regard the forward as one of their top targets for the summer after winning promotion to the Premier League.
However, according to The Sun, new Hull boss Leonid Slutsky is ready to lodge an offer for the attacker as he prepares the Tigers for their first season back in the Championship.
The Blues are likely to want Abraham to gain experience in the top flight, but Slutsky is looking to take advantage of an emerging link between the two clubs due to his friendship with Chelsea owner Roman Abramovich.
The Russian billionaire allegedly played a role in Slutsky being given his opportunity in England, and the former CSKA Moscow coach has openly acknowledged that he may look to sign some of Chelsea's young stars.
Abraham netted 26 goals in 48 appearances for Bristol City during the most recent campaign.