Meyer will become a free agent at the end of this season, and the Daily Star's David Woods reported Gunners manager Arsene Wenger "is already working hard" to beat Liverpool to the German's signature.
Wenger already has two Schalke alumni among his ranks in Mesut Ozil and Sead Kolasinac, and Meyer is attracting major attention as he approaches the end of his contract.
The 22-year-old has enjoyed a resurgence since converting to a more disciplined holding role under manager Domenico Tedesco, and Tifo Football recently explored Meyer's rise back to his best at the Veltins-Arena:
Bayern have already secured the signature of one Schalke star in Leon Goretzka, who will join the club when his deal runs out this summer, and they'll naturally be considered competition to land another German prospect.
Football Radars posted in December to show how Meyer's passing presence in particular has been emphasised this season:
Woods' report also included recent quotes given by Schalke executive Christian Heidel, who said of his player as he enters the final few months of his contract:
"Max will be deciding shortly if he wants to stay at Schalke past the summer.
"Max is very happy here, Schalke is his home, we will try to make it him as comfortable financially as possible.
"He is playing the best season of his life, and the coach has found a position for him that Max didn't believe he could play.
"Personally, I didn't think he could either."
Barca have been speculated as another interested party, and Catalan daily Sport cited the Daily Star in reporting Philippe Coutinho's signing wouldn't affect their pursuit of Meyer.
After spending so heavily on the signings of Coutinho and Ousmane Dembele in the past two transfer windows, it's free pickups like this that Barcelona may find most attractive.
Arsenal signed Granit Xhaka in the summer of 2016 but are still seeking an assured engine-room presence, while Liverpool have RB Leipzig star Naby Keita ready to arrive in the summer after agreeing a deal in 2017.
One Gunners fan speaking on BBC 5 live's 606 recently said the north Londoners have been missing a settled defensive-midfield star for going on a decade:
The low-cost nature of Meyer's prospective purchase means Arsenal would be more able to compete with the rest of the market in terms of the financial package they could offer.
One of the Bundesliga's most revered up-and-comers looks like he will change his surroundings this summer, and two of the Premier League's elite appear set to battle for his signature.