Martial's agent Philippe Lamboley confirmed on Friday that his client is keen to leave Old Trafford during the January transfer window, having slipped down the pecking order since the arrivals of Cristiano Ronaldo and Jadon Sancho.
The 26-year-old has been used sparingly so far this season, making only two Premier League starts and four across all competitions.
A knee injury has so far scuppered his chances of impressing new interim boss Ralf Rangnick, and Martial may have already played his final game for the club if Man United do opt to offload him when the window reopens next month.
The Magpies are looking for a marquee signing to help drag them away from relegation danger following their takeover, while Arsenal are in the market for attacking reinforcements amid uncertainty over the futures of Pierre-Emerick Aubameyang, Alexandre Lacazette, Eddie Nketiah and Folarin Balogun.
Barcelona, meanwhile, are also looking to bolster their attacking options following the news that summer signing Sergio Aguero may be forced into early retirement with a heart issue.
PSG are thought to be the least likely destination for the winger, although they could increase their interest if Kylian Mbappe leaves for Real Madrid.
Martial has spent six-and-a-half years at Manchester United, scoring 79 goals in 268 appearances.