The Premier League side have reportedly been monitoring the 18-year-old for much of the season.
The Red Devils have been linked with the midfielder for several months, and The Sun suggests that a bid may be placed in the summer as they attempt to see off competition from Bayern Munich, Barcelona and Real Madrid.
But The Sun claims that United can increase their chances of landing Sanches if they replace Louis van Gaal with the former Chelsea boss at the end of the season as both the 18-year-old and Mourinho are represented by the same agent, Jorge Mendes.
Having only made his first-team debut in October, Sanches has clearly taken little time to put himself on the radar of several of Europe's biggest clubs. He has scored two goals from his 14 league appearances for the Portuguese giants, including the winner against Vitoria at the start of January, to help put them at the top of the table.
And it appears that United are potentially hoping that the link between Mourinho and Sanches would enable the Special One to get the very best out of the exciting youngster if they were to both move to Old Trafford at some point in the future.