Jose Mourinho is keen to beat former club Real Madrid to the signing, but the Spanish giants are reportedly preparing a £50m offer of their own.
Cavani is Chelsea's top transfer priority this summer after they missed out on Monaco-bound Radamel Falcao, and the Stamford Bridge club hoped they could tempt Napoli and Rafa Benitez with the offer of Fernando Torres.
But the Serie 'A' club would rather a straight cash deal, preferably for the striker's full £53m minimum fee release clause.
And Chelsea face a real fight to land one of the hottest properties in world football, as the report claims Real Madrid have also made Cavani their number one transfer target ahead of Liverpool's Luis Suarez.
And they could steal a march on Chelsea as Mourinho confirmed nothing would be decided regarding Torres until the forward returned from international duty at the Confederations Cup.
That gives Real Madrid the opportunity to wrap up a deal, and the report claims they're already in talks with the player's representatives.
Manchester City were another linked with a move for Cavani, and the report suggests Napoli turned down one bid from the 2011-12 Premier League champions - believed to be a £25.5m plus Edin Dzeko offer.
Cavani is playing his cards close to his chest, and insisted last weekend he'd won't talk about his future until after the Confederations Cup.
"Everything is uncertain at the moment," Cavani told OMNISPORT. "Today I have a contract with Napoli, but on the other hand there are some conversations going on and now things are a bit dynamic.
"What I know is that I belong to Napoli and that I'm thinking about my national team because we have a very difficult qualifying game away to Venezuela and we also have a difficult Confederations Cup tournament to deal with.
"After all these commitments I'll think about what is going to happen."