The German international's current deal at Veltins-Arena will run down after the end of this campaign. The Bundesliga outfit are keen on retaining him and have already proposed a club-record contract of €12m-a-year (£10.4m, $14.1m).
Barcelona are also believed to be in the race for the signature, while Champions League winners Real Madrid are also interested in signing him. According to The Sun, a host of European clubs are keen on securing his services and the list includes the likes of United, City and Arsenal.
However, the north London club have managed to beat the Red Devils, Pep Guardiola's side and the Catalan club for the midfielder's signature. The 22-year-old is expected to join Arsenal as a free agent after the end of this term.
Goretzka's ability to play anywhere in the midfield has seen him attract interest from the top clubs in Europe. Arsenal manager Arsene Wenger has been following his progress, having the Gunners' scouts watch him over the last few years.
Despite interest from other clubs, Arsenal are believed they have the edge over the likes of United, City, Barcelona and Real in signing the Schalke star. The Emirates club are willing to pay a hefty signing-on bonus to Goretzka.
Schalke CEO Christian Heidel earlier admitted the German outfit will not sanction their star player's sale in the January transfer window.
"Although we have set no time, I believe that Leon will decide in the winter. It seems to be so that other clubs are interested and they will not wait until June 29 [when Goretzka's contract expires]," Heidel said, as quoted by Ruhr Nachrichten.
"The positive development the team is currently undergoing will certainly not worsen our situation."
Meanwhile, Goretzka has also publicly revealed that he will make a decision on his future in January.
"I have to find out for myself what the next step should be. I have to consider a million things. I am a person who is very meticulous in such decisions. I play through all the scenarios. So I need time," the player explained, as relayed by Goal.com.
"There doesn't exist a deadline, but I think we are in the final phase. The final decision will be made in January."
Goretzka joined Schalke from Bochum in 2013 and it is likely that his association with his current employers is all set to be over after the end of the season. Should Arsenal reach a pre-contract agreement with the German midfielder, it will be a massive boost for the Gunners.
Wenger is expected to lose Mesut Ozil and Alexis Sanchez on a free transfer after this season. The duo's current deal at the Emirates will expire in 2018 and are attracting interest from United and City respectively.