The People says United's new manager Louis van Gaal and Arsene Wenger, his counterpart at the Gunners, are both after the former Spanish international, who is now a free agent.
Valdes is available after his move to Monaco – and a £25million a year tax-free contract – fell apart when he failed a medical.
The 32-year-old damaged cruciate ligaments last season.
Van Gaal worked with Valdes when in charge at Barca. Sources claim the Dutchman is concerned David de Gea – who was excellent last season – has no rival for the No.1 spot.
However, Arsenal also want competition for Wojciech Szczesny and have asked Valdes to come over for talks.
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