According to Neil Fissler in the Sunday Express, the Blues have been scouting the central midfielder and plan to make a move to sign him in January, while Barcelona have been tracking him for more than a year.
He has yet to make an appearance for Bayer's senior side, but he is a Germany youth international and has captained their under-18 side in the past.
Akkaynak, 18, has a contract with Leverkusen that expires in 2019, so the Bundesliga outfit will need to move quickly to sign him to new terms if they want to keep him at the club.
They would also likely do well to hand him a senior debut soon and promise him regular game time or he may be tempted elsewhere.
Chelsea and Barca are both huge clubs, and it would be no surprise if Akkaynak were tempted by a potential move to Stamford Bridge or the Camp Nou, respectively.
But it is likely he will be afforded more opportunities at first-team level if he stays at the BayArena and competes in the Bundesliga.
Akkaynak is a central midfielder capable of both playing in a more advanced role or lying deeper to protect the defence.
Chelsea sold Nemanja Matic from their engine room in the recent summer transfer window, but they also brought in Tiemoue Bakayoko and Danny Drinkwater.
Given the continued presence of central midfielders N'Golo Kante and Cesc Fabregas at Chelsea, it is not likely there would be an immediate place for Akkaynak in the first-team squad, even with the Blues back in European competition in 2017-18.