The Premier League champion has made its move for the teenage prodigy as the club attempts to make a second statement signing of the January window.
United wants to follow up the £37 million capture of Juan Mata by trumping Chelsea to the signing of Shaw.
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Shaw, 18, is the highest rated English player in his age group. He has yet to be capped by England but made his debut for Southampton aged 18 and has already accumulated 49 first-team appearances for Mauricio Pochettino's improved team.
Shaw signed a new five-year contract with the Saints last summer but sources have said that it is "possible" he could leave the club in the final six days of the window.
Goal revealed in midweek that Moyes was looking to add two players to his squad after tying up a deal for Mata, who will join from Chelsea for a club-record fee subject to completing a medical on Saturday.
United is now stepping up its efforts to secure Shaw, who has also attracted strong interest from Chelsea.
The Londoners could now go head-to-head with United for Shaw, as Ashley Cole has lost his regular starting spot at Chelsea in recent months to Cesar Azpilicueta and the Englishman, 33, is out of contract in the summer.
Moyes has turned his attention to Shaw after failing to clinch deals last summer for Everton's Leighton Baines and Real Madrid's Fabio Coentrao.