The 23-year-old has caught the eye with his impressive performances during the 2021-22 season, finding the back of the net on 17 occasions in 36 appearances across all competitions in addition to contributing 13 assists.
Gakpo, who helped PSV win the KNVB-Beker and qualify for the Champions League last term, still has four years left on his contract with the Eredivisie giants.
United boss Erik ten Hag is familiar with the Dutchman from his time in charge of Ajax and is said to approve of a move for the Netherlands international.
Manchester United were initially hopeful of signing Antony from Ajax after the Brazilian became a regular for the Dutch champions under Ten Hag.
However, Ajax are not prepared to let him leave for anything less than £80m this summer, which has forced Ten Hag to rethink his attacking targets.