TNT Sports says United are looking at Martinez as a possible replacement for De Gea.
Juventus are also in the market for a keeper after it was announced this week that 43-year-old Gianluigi Buffon will be leaving the club this summer.
But if United were to win the race for Martinez, it would not be a good look for Arsenal.
Leno is now linked with a move away from the Emirates after making a number of high-profile errors this season.
The Athletic claimed this week that Leno is one of five senior Arsenal players who want to leave the club.