The Blues will however, have to pay £9.1 million for the experienced striker who joined the Yellow Canaries on a three-year deal from Manchester United over the summer.
Fenerbahçe signed Van Persie for £4.2 million but he only had a year remaining on his contract at the time.
Chelsea have been struggling upfront this season with first choice striker Diego Costa netting just twice and on-loan forward Falcao having scored just once all season.
Barcelona have also been closely linked with a move for the Netherlands international star in an attempt to add depth to their forward line over the winter transfer window.
Van Persie already has five goals and an assist to his name season but is rumoured to have had a rift with head coach Vitor Pereira although both have officially denied such speculation.