The report claims United have informed Southampton of their continued interest in the England under-21 international.
They reportedly had a £22.5m bid for Shaw turned down in the January transfer window, but United boss David Moyes remains keen to bring the Chelsea target to Old Trafford.
Shaw is a Chelsea fan, and the Blues have often been linked with the left-back when his name is mentioned in transfer gossip columns.
Southampton, despite the upheaval following the departure of executive chairman Nicola Cortese, held onto Shaw through the winter window.
But United remain keen to do a deal for the talented 18-year-old this summer.
The report claims United have told Southampton their bid for Shaw "remains on the table."
Shaw has been tipped by many as an outsider to make Roy Hodgson's England squad for the World Cup in Brazil.
Hodgson has confirmed he will get a chance to impress in a friendly next month, although he faces stiff competition for a place at the left-back position.
Still, the Chelsea and United target has been one of the most consistent full-backs in the league this term.
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