The 17-year-old left-back first appeared on Moyes' radar earlier this season, when the then-Toffees manager was looking to provide first-choice full-back Leighton Baines with an understudy.
Baines, who had been strongly linked with Manchester United, would have then been replaced by Shaw at Goodison Park next term.
The report claims Moyes will now look to make Shaw one of his first United signings, as he looks for a long-term alternative to Patrice Evra.
Shaw is yet to sign his first senior contract at Southampton, but has pledged his future to the south coast club, and intends to do so on his 18th birthday in July this year.
The boyhood Chelsea fan has been linked with a move to Stamford Bridge, but the managerial uncertainty at the club may put the teenager off for the time being. He's been reportedly earmarked as a long-term replacement for Ashley Cole.
He is also getting regular first-team football under Mauricio Pochettino, and a report in the Daily Telegraph claims he's eager to stay at Southampton this summer despite interest from Chelsea.
Shaw is not the only England youth team star to interest Moyes, though. The report claims the Glasweigen may return to Goodison Park to bring 19-year-old midfielder Ross Barkley to Old Trafford.
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