The Daily Express says the Blues have sent scouts to watch the young German in action and could make a bid for him in January.
But they face competition from La Liga giants Barca, who have been keeping an eye on Akkaynak, 18, development for over a year.
And the report suggests that Leverkusen could be interested in cashing in on Akkaynak with his current contract running until 2019.
Akkaynak is yet to play for the first team but that has not stopped some of Europe's elite from watching his progress.
He is able to play in both defensive and attacking midfield which gives him great promise to other teams.