The 22-year-old is out of contract at the end of the season and has been linked with a move to a number of top European sides, however, according to 'Sky Sports News' United have now joined the race for the player.
The German's contract status means that he can open talks over a pre-contract with foreign sides on January 1, something which has made him highly sought-after.
Schalke are desperate to keep their star man though and are thought to have offered him a contract that would see him become the highest earner in the club's history.
Goretzka is a classy, yet versatile midfielder, capable of dictacting play from deep, but also from chipping in with goals at the other end, as shown by his impressive record of six goals in 12 games for Germany.
He is considered to be one of the brightest talents in German football and has been tipped to shine for 'Die Mannschaft' at next summer's World Cup in Russia.
A number of sides have been linked with a move for the youngster, with Barcelona, Juventus, Tottenham and Arsenal all thought to be seriously considering a swoop.
United may be in the market for a central midfielder come the summer, with Belgian Marouane Fellaini looking more and more like leaving for pastures new when his contract comes to an end at the end of the season and Goretzka would certainly fit the profile that Jose Mourinho's side would be looking for.