Goretzka, 22, is in the final year of his contract with the Bundesliga side with his possible availability next summer set to spark a battle for his signature. Bayern Munich has been strongly linked with a move for the Germany international but clubs from elsewhere in Europe are eager to try and tempt him away from his homeland.
The Gunners are facing the very real possibility of losing Mesut Ozil on a free transfer next summer, possibly to United, and Goretzka, who can operate in the former Real Madrid star's number 10 position, could be seen as an ideal replacement.
Liverpool manager Jurgen Klopp meanwhile was rumoured to have met up with the player during the summer transfer window to discuss a move to Merseyside.
The Premier League trio may also have a slightly clearer path in their attempts to sign the player with Barcelona seemingly not ready to follow up their interest.
The Catalan giants were said to be keen on wrapping up a deal before Bayern Munich get their hands on another German talent, according to Mundo Deportivo.
Reports had claimed Barcelona might not even wait until next summer when Goretzka will be available on a free transfer and will instead strike a cut-price deal with Schalke in January to try and beat Bayern to the punch.
Barcelona's chief international scout Ariedo Braida was in attendance as Schalke drew 1-1 with Bayer Leverkusen last Friday [29 September] – with Goretzka scoring his side's goal with a well-struck free-kick.
But while he has been impressed with what he has seen from Goretzka, a move this January could be premature.
Asked about the Germany international, Braida said: "Without a doubt he is a great player. But it is too soon for Barcelona," SPORT report.