The 18-year-old playmaker has also attracted interest from the likes of Manchester United, Juventus and Real Madrid, but the report claims the Gunners and Gladbach now lead the race.
The teenage starlet has been compared to Bayern Munich-bound Germany international Mario Gotze after a series of stunning performances in the German second division, Bundesliga 2.
Reports in Germany claim Goretzka could command a fee in the region of £3m, but he'd fit the profile of an Arsenal signing - young, full of potential, and with re-sale value.
And Goretzka's manager at VfL Bochum, Walter Schneck, has heaped praise on the teenager, tipping him to become one of the world's top players.
"I have never seen an 18-year-old who has so much class and desire," said Schneck as reported by the Daily Express.
"The boy will find his way into the world's best."
Goretzka is a Germany under-19 international, has captained his country at previous age groups, and has made over 30 league appearances for Bochum since breaking into the first-team last year.
Both Arsenal and Gladbach have a good record of turning promising players into world-class stars. Most recently, Gladbach brought Marco Reus through from their youth team to first-team, eventually selling him to Borussia Dortmund for €17m.