Goretzka, 22, is in the final year of his contract at the Bundesliga club and will be able to leave on a free transfer at the end of the 2017-18 season.
Bayern are said to be favourites to keep the Germany international in his homeland but Barcelona and United are hoping to tempt him away, while both Arsenal and Liverpool are also among the interested parties, according to Bild.
With such attractive offers soon to be on the table, Schalke's hopes of holding onto the midfielder would appear slim, with their skipper Fährmann admitting it will be difficult to blame the player for wanting to move onto bigger and better things.
He told Reviersport: "I would like him to stay here for a long time. But you also have to be honest, a man with his qualities is hard to keep in the long run and he will end up in the top clubs in the long term, the question is only when."
Barcelona 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 the Catalan giants might not 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.
But recent comments from Barca scout Ariedo Braida would suggest the club might hold fire for the time being. "Without a doubt he is a great player. But it is too soon for Barcelona," Braida commented after travelling to Germany to watch Goretzka in action on 29 September.
Barcelona's interest remains widely reported in both Germany and Spain, however, but anyone wishing to sign the talented midfielder may have to wait until the summer. A recent report from Sport Bild suggests Schalke will not consider cashing in on the player while they still can in January because they would lose up to 40% cent of any potential fee due to various clauses inserted in his contract.
Goretzka has scored twice and provided one assist in eight matches for Schalke this season having notched for his side last season.