The Italian midfielder could be on his way out of the Stadio Olimpico, and Goal claim Roma will listen to offers above £15 million.
Manchester City were in pursuit of De Rossi last summer before he announced his intentions to stay put - however now Chelsea appear to be in the driving seat for his signature.
Despite that Goal are now reporting that both Real Madrid and PSG have made their interest known to Roma and could challenge the Blues.
De Rossi reportedly held talks with sporting director Walter Sabatini following his return from the Confederations Cup with Italy, and although new boss Rudi Garcia would like to keep him the club is open to selling up to the likes of Chelsea or Madrid.
Goal claim Chelsea held preliminary talks with the 30-year-old back in the winter, and Jose Mourinho is said to be an admirer of his ability.
Chelsea may need to be swift given the interest of two European giants if they want to secure to services of De Rossi, who would provide competition for the likes of Frank Lampard, John Obi-Mikel and newly-signed Marco van Ginkel.