But the defending Premier League champions are not the only team interested in the forward, with Inter Milan and Paris St Germain planning approaches of their own.
Spanish paper Sport claims United are thought to be considering a January swoop for the Chilean international as a replacement for Nani and Antonio Valencia.
Nani has been linked with a move away from Old Trafford, while Valencia - like fellow United winger Ashley Young - has been struggling for form this season.
Moyes is believed to be keen to strengthen his midfield options in the winter window, and will monitor Sanchez in the coming weeks ahead of the New Year.
But a separate report in AS claims that new Inter Milan owner Erick Thohir is set to hand manager Walter Mazzarri a bumper transfer budget - with the Barcelona forward at the top of his wish list.
Also followed by PSG, Sanchez has enjoyed a break-out season at the Nou Camp this term.
He initially struggled to settle following his move from Serie A, but seven goals in nine La Liga appearances this season - including a stunning chip in El Clasico - have established the former-Udinese star as a fixture in Gerardo Martino's side.
United have several options out wide, including Nani, Valencia, Young and Wilfried Zaha but none of the quartet have staked a claim for a regular berth so far under Moyes.