Lingard is expected to join West Ham United as Everton lose out, according to the Daily Mail.
Lingard, 29, is a free agent whose contract with United expires on June 30, but has received a number of offers. Lampard has identified the England hitman as one of Everton's summer targets, with Newcastle and Tottenham also showing interest.
But Lingard is expected to reject those clubs in favour of a return to Moyes' side West Ham United.
Lingard was on loan from Manchester United at West Ham United in the second half of the 2020-2021 season, when he scored nine goals and five assists in 16 appearances. But on his return to Manchester United, he still didn't get a chance, picking up just four starts in all competitions last season.
Lingard is hoping to get regular playing time for his next team and is keen to return to the Three Lions in time for the World Cup.
West Ham United are considering other midfield options after their ￡20m bid for Amadou onana was rejected by Lille OSC.