Cillessen has spent the last two years at Barcelona in the shadow of Marc-Andre Ter Stegen.
And, unwilling to sit on the bench for much longer, Barcelona expect him to go.
However, Mundo Deportivo say another name on Sarri’s list is Inter Milan ace Samir Handanovic.
The shot-stopper is entering the twilight years of his career.
But Chelsea may pounce to bring him to the capital.
Meanwhile, Sky Sports man Rob Wotton has provided an update on Liverpool’s Alisson pursuit.
Chelsea transfer news: Alisson Becker is close to joining Liverpool (Image: GETTY)
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"Liverpool and Chelsea have been interested in Roma goalkeeper Alisson. It seems both clubs could be battling it out for the signing," Wotton told Sky Sports' Transfer Centre show.
"Sky in Italy are reporting that Italy have made a £62million bid although we understand a fee is yet to be agreed.
"Sky Sources are telling us that Liverpool believe there is potential to do a deal although the club aren't commenting.
"We were told earlier in the window that Liverpool believed Alisson was moving to Real Madrid.
"Sky in Italy are saying Roma will sell and Chelsea are willing to turn their interest into a formal offer."