Moyes tried to sign the England star, 22, on loan when he was in charge at Everton last season.
A source told The Sun: "Andros was one of Everton's summer targets before Moyes left.
"And the impact Andros has had in the first half of this season — establishing himself at Spurs and making an instant impact for England — has not gone unnoticed.
"There is a view that he is seen as a longer-term replacement for the likes of Antonio Valencia and Nani on United's flanks."
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