The silky attacker is expected to leave Barcelona this summer for around £25m despite his heroics in Brazil, with the La Liga giants plotting to overhaul their first-team.
Sanchez starred for his country as they were knocked out on penalties by the host nation, and although he missed his spot kick, he netted a vital goal in normal time.
Liverpool have been heavily linked with the 25-year-old in recent times as they plot for the potential loss of Luis Suarez to Barca, but now United are ready to beat them to him.
Louis van Gaal has already moved to splash the cash for Luke Shaw and Ander Herrera, but he is not yet content with his inherited Old Trafford squad.
Sanchez is seen as the ideal man to add to some spark in the final third to get the best out of Robin van Persie and England World Cup flop Wayne Rooney.