The 33-year-old is currently with Chinese side Shanghai Shenhua after joining from Chelsea in January 2011, but has since become unhappy with life in Asia.
And rumours of a switch to les Parisiens have gathered momentum after the striker began training with the club, though neither the player or coach Carlo Ancelotti have commented on the growing speculation.
The former France international played for PSG as a teenager in 1996 before rejoining them in 2000. He has also had spells with Liverpool, Bolton, Fenerbahce and Manchester City.
Anelka, who plays alongside his former Chelsea strike-partner Didier Drogba at Shanghai, has scored three times for his current side in 24 matches.