Antonio Conte’s side have been heavily linked with the former after the Belgian striker said he wasn’t planning to sign a contract extension with the club.
But the Mirror say the Blues are also considering a move for midfielder Barkley as well.
The England international will have 12 months to run on his contract this summer.
And that could mean he’s on the way out of Goodison Park if both parties can’t work out an extension.
The Mirror say Chelsea would have to splash out £100m to land the pair.
But they could have a chance to recoup their money if striker Diego Costa or goalkeeper Thibaut Courtois leave in the summer window.
Costa has been linked with a move back to former club Atletico Madrid.
And Real Madrid are thought to be eyeing up a move for Belgian ace Courtois.