The Chilean is currently in the last 18 months of his contract at the Emirates and it remains unclear whether or not he will sign a new deal with the club.
Sanchez joined the Gunners from Barcelona in the summer of 2014 and has become a key member of Arsene Wenger's side.
However, PSG are keen to take advantage of his situation and believe they may be able to snatch him for a cut-price fee in the summer should he not sign a new deal in north London.
Director of football Patrick Kluivert is believed to be a huge fan of the 28-year-old and it is thought Les Parisiens will be able to offer more than the reported £200,000-a-week Arsenal are prepared to pay him.
They could face competition from elsewhere, with clubs in China also tracking the player.