According to Paul Hirst of the Times, the Chilean is far from content with life at Old Trafford despite only joining the Red Devils in the January transfer window on a four-year deal worth around £80 million.
It's suggested Sanchez has struggled to settle in the north-west since his switch from Arsenal, and his agent, Fernando Felicevich, "has started looking at potential suitors for his client, and PSG have made it clear that they would be interested in him, although it remains to be seen whether United would sell."
According to Hirst, there were discussions between Sanchez's representatives and PSG in the summer of 2017, although the French giants eventually opted to sign Kylian Mbappe instead.
United boss Jose Mourinho is reportedly doing what he can to get the best out of Sanchez, although he's described the drastic loss of form from the forward as a "mystery" to members of his staff.
After all, before the Red Devils brought the Chile international to the club, he was one of the most feared forwards in the Premier League, excelling in his three-and-a-half years at Arsenal.
Arsene Wenger, who was key in getting the best out of Sanchez with the Gunners during his time as manager, offered his thoughts on the player to beIN Sports:
beIN SPORTS @beINSPORTS
Arsène Wenger was special guest on our Champions League coverage this week and discussed: 🔴 His Arsenal mistakes 🔥 Neymar 🏅 His Ballon d'Or winners ⁉️ Alexis Sánchez 🔮 His future ... and Cristiano Ronaldo's six-pack 😂 Fascinating as ever from 'Le Professeur' 👨🏫 https://t.co/Ba9XiFRNKl
Since moving to the Red Devils, the ex-Barcelona man has mustered only four goals in 28 appearances. In 2018-19, he's netted once, albeit that was a crucial winner in the 3-2 comeback win over Newcastle United on October 6.
Even so, Sanchez has looked nowhere near the force of nature he was at his best for Arsenal. WhoScored.com summed up how effective he was in the final third for his former club:
Alexis Sanchez: Since moving to the Premier League in 2014, only Sergio Aguero (76), Romelu Lukaku (75) and Harry Kane (74) have had a direct hand in more goals from open play than @Alexis_Sanchez (70 -- 48 goals, 22 assists) Full player stats -- https://t.co/ojSfRKiTBQ https://t.co/cCDbYr82wn
At United, those sorts of moments have been few and far between for Sanchez, and it'd be intriguing to see what the Red Devils would do if they got a decent offer for him in January.
The prospect of Sanchez moving to PSG would be a fascinating one. After all, they already have Mbappe, Neymar and Edinson Cavani at their disposal. The former duo have been especially potent in front of goal lately:
Scouted Football @ScoutedFtbl
10+ direct goal involvements in Europe's top five leagues this season: 13 • Kylian Mbappé (19) 12 • Lionel Messi 12 • Neymar 12 • Nicolas Pépé (23) 11 • Sébastien Haller 11 • Cristiano Ronaldo 11 • Luis Suárez 10 • Eden Hazard 10 • Jadon Sancho (18) 10 • Suso Elite. https://t.co/yBtRv8BTbW
Sanchez, at his best, would be another thrilling attacker to add to the setup, and he could potentially step in to lead the line in place of Cavani on occasions.
Still, he'd be an expensive acquisition in terms of a transfer fee and the salary he'd likely command. With that mind, it's tough to see a January move materialising. Sanchez's best chance of playing a part in a successful team may be taking his chances when they do arise at United.