According to John Cross of the Daily Mirror, with less than a year remaining on the Chilean's contract, the French giants are said to be keen to capitalise, with the Gunners considering selling their star man.
Cross suggested that if PSG do want Sanchez, they may have to move in midseason, with Manchester City showing serious interest in the player.
"PSG have again been in contact with the Chilean's representatives, and he would be another massive addition to their star-studded squad," he wrote. "They also believe the added attraction of moving to Paris from similarly cosmopolitan London could be significant in beating their Manchester rivals to his signature."
The situation is bleak for Arsenal, with Cross noting the Gunners are resigned to losing the 28-year-old, with no movement on a new deal.
If he were to exit the club, there would be a massive void in the final third. As we can see, his team-mates rate him highly:
Sanchez is one of the finest players in the Premier League, and since arriving at the Emirates Stadium in the summer of 2014, he has consistently led the charge for Arsenal.
The former Barcelona star brings vibrancy to an attack. Whether deployed on either flank or through the middle, defenders find him difficult to pin down. Sanchez is always on the move, asks questions of opponents and tends to make smart decisions with the ball.
As BT Sport Score noted, Sanchez is a productive player in terms of the goals he scores and makes:
While the prospect of the Chilean joining City is ominous for the rest of the Premier League, seeing him move to the Parc des Princes would be scary for the rest of European football.
After all, PSG are already among the most fearsome attacking sides in the game. The summer additions of Neymar and Kylian Mbappe have provided some fresh impetus in support of poacher Edinson Cavani.
The results have been spectacular, with PSG surging to the top of the Ligue 1 table and winning all of their UEFA Champions League games. Per The Sportsman, the trio has been incredible in front of goal:
For Sanchez, the chance to link up with City manager Pep Guardiola for the second time in his career would be an exciting opportunity, especially given the way the Sky Blues have coasted to the top of the Premier League table this season.
Still, there's something special going on in Paris, and it is easy to see the Chilean being enticed by their project. And while Arsenal would prefer to keep Sanchez at the club for many years, if he is to depart, they would surely prefer to see him head abroad instead of to a domestic rival.