The 28-year-old Argentine has impressed for the Villains since his arrival in September, keeping 14 clean sheets in the English top flight.
According to Football Insider, the Red Devils are planning to let both David de Gea and Sergio Romero leave the club as part of a major overhaul, with Martinez one of their prime targets.
The 20-time English champions are also believed to be keeping close tabs on Atletico Madrid’s Jan Oblak, AC Milan’s Gianluigi Donnarumma and Burnley’s Nick Pope.
United head coach Ole Gunnar Solskjaer would like to bring in two senior goalkeepers to strengthen his options and provide more competition to Dean Henderson.
Martinez has enjoyed an impressive 2020-21 campaign, helping the Villains climb up to ninth in the table.