But the Premier League trio will reportedly have to stump up at least €70m to pry him from the Gelsenkirchen club before the World Cup.
The 20-year-old has made more than 75 Bundesliga appearances for Schalke since his debut in 2011, and his impressive performances in the Champions League have convinced the likes of United and Arsenal to firm up their interest.
Draxler was one of Schalke's best players in their two defeats to Chelsea in the Champions League, and the Germany international has been strongly linked with a move.
Earlier this week, reports claimed Arsenal boss Arsene Wenger had made contact with the player over a future move.
The Gunners are well stocked in the position, with Mesut Ozil, Santi Cazorla, Jack Wilshere and Aaron Ramsey all competing for playing time.
But Chelsea are in a similar situation. In fact, Jose Mourinho's side may actually have too many options in attacking midfield as the likes of Juan Mata cannot get a regular run in the team.
That leaves Manchester United, with David Moyes expected to strengthen his midfield in January despite the addition of Marouane Fellaini over the summer.
But the prohibitive price - Schalke are reportedly demanding €70m - may be too steep for the Red Devils. Having said that, reports earlier this week suggested Draxler could be available for half that fee.