The France international is currently playing in Greece with AEK Athens, having made the move almost two years ago.
He made 23 appearances for the club last season, scoring nine goals and providing two assists, as they eventually finished fourth to qualify for the Conference League.
Now local Mexican journalist Cesar Luis Merlo is reporting Pumas want to bring him to Mexico, with the Liga MX transfer window closing on Friday.
Merlo stated on his YouTube channel: "AEK is willing to loan him to Pumas free of charge (they should only pay his salary), but Martial earns a gigantic salary, gigantic.
"They've been talking to him for three or four days, but the player still hasn't given in to his demands.
‘I'm not ruling him out 100 percent, but if he hasn't budged in four days, I don't see it being easy for him to move today.
"The reality is that Pumas is trying to get Martial, but the player doesn't want to budge even a euro.
"For now, neither Martial nor his agent have budged. If they do, there may be a chance, but we'll have to wait and see."