The 25-year-old has scored nine goals so far this season, having signed a contract until June 2015 with the Bundesliga side last summer.
Borussia Dortmund, Wolfsburg and Lazio have previously been linked with Szalai, but German and Hungarian media have reported that the Gunners have been monitoring his progress.
However Mainz, currently sixth in the Bundesliga, are reluctant to part company with the striker just yet.
"Arsenal haven't contacted us," claimed Mainz manager Christian Heidel. "We don't want to let our striker go in the winter transfer window. But that's another situation if someone willing to pay 20 million Euros for him."
Szalai himself is in the dark over any interest from Arsenal, recently telling Kicker: "I don't know anything about it. The season is long, I don't know what will happen at the end of it.
"My aim is to collect as many points as we can and get as far as we can in the German Cup. Summer we can talk about what happened."