Milners current contract expires this summer, and the 33-year-old is yet to agree on fresh terms to extend his stay at Anfield.
As a result, Le Parisien claim that PSG are ready to push for a free transfer this summer, keen for Milner to follow Manchester United star Ander Herrera through the door at the Parc des Princes.
He is currently free to negotiate a contract with overseas sides, although it is expected that he will wait until after the upcoming Champions League final against Tottenham Hotspur, which could be Milners final game for Liverpool.
Midfield has been a real problem for Thomas Tuchels side this season, and they could find themselves on the lookout for a number of reinforcements. The report claims that Leandro Paredes, Julian Draxler and Christopher Nkunku are all nearing the exit door at the club, whilst Adrien Rabiot will make his long-awaited departure from PSG in the coming weeks.
The club are keen to raise money to appease UEFAs Financial Fair Play regulations, meaning they are on the lookout for free transfers. Herreras switch to Paris is expected to be confirmed in the near future, and Milner could prove to be another intriguing option.
The 33-year-old has been a vital part of Liverpools side this season, making 44 appearances in all competitions. He has operated primarily as a central midfielder, although has filled in at full-back on a number of occasions, proving his worth to Jurgen Klopp.
However, despite this versatility, there remains a huge amount of speculation around his future. His last contract extension came in 2015, and there appear to be doubts over whether he will agree on a new deal to remain at Anfield for at least another year.