The Red Devils are known to be in the market for a new left-back this month with Patrice Evra’s contract set to expire at the end of the season.
Moreno was believed to be one of Moyes’ top targets, but it is now understood that the Scot has dropped his interest with the La Liga side unwilling to discuss a deal.
The ex-Everton chief will instead focus on English talent, with Shaw one of his favoured options.
The 18-year-old ace has been, arguably, the finest young defender in the Premier League over the course of the last 12 months, establishing himself as a key member of the Saints’ impressive first-team unit.
Despite claims that the South Coast club are unwilling to let their teenage sensation leave, Moyes is prepared to make a serious effort to lure him to Old Trafford.
Even with his youthful age, Shaw is valued at around £20m, and is also wanted by Chelsea.
Leighton Baines is another players on United’s radar, but Everton are reluctant to do business with the Premier League champions after being forced into selling them Marouane Fellaini late on in the summer transfer window.