A number of reports today linked United with a £30m bid for the Southampton left-back, with the Old Trafford club ready to offer him £80,000-a-week and the opportunity to replace Patrice Evra.
And the Daily Mail claims once that's done, Moyes will get straight onto tying up the deal for his second top target of the summer, William Carvalho.
The 21-year-old Sporting Lisbon midfielder has been one of the stand-out players in the Portuguese top flight this season, and United have scouted him extensively.
The report claims United opened preliminary talks with United in March, and are working to reduce Sporting's asking price.
The Portuguese club want Carvalho's full £37m release clause, but United, who may have to pay up to £30m to land Shaw, are reluctant to go that high.
Shaw, who became an England international earlier this season, remains the number one target for Moyes. But the fact that the Carvalho deal is moving forward as well is evidence of the United manager wanting to avoid the mistakes of last summer.
Several ill-fated pursuits late in the summer window made a mockery of Moyes and United's transfer plans, as they were forced to move for Marouane Fellaini on deadline day.