Arsenal boss Arsene Wenger is known to be a long-term admirer of the Germany international, but missed out on signing him when he joined PSG from Wolfsburg last year.
However, the 24-year-old has struggled to pin down a regular starting place in the PSG side since his arrival, and his first-team chances could become even more limited should PSG go in for Arsenal's Alexis Sanchez during the January transfer window.
The Gunners could look to take Draxler in the opposite direction once the window reopens next month, though, and The Mirror reports that they have now been given a £40m asking price.
Any deal for Draxler would likely be financed mostly by an exit for Sanchez, with Arsenal considering cashing in on the Chilean rather than losing him on a free transfer at the end of the season.
Draxler has started just 10 Ligue 1 matches so far this season, scoring two goals and creating three more.