The 22-year-old midfielder is out of contract at the end of the season.
He is wanted by a string of Euro giants including Barcelona and Bayern Munich.
But Arsenal boss Arsene Wenger has stepped up his interest.
According to the Mirror, one of Wenger’s most trusted spies has been closely following Goretzka.
The scout has watched Schalke previous three Bundesliga games.
Arsenal are also willing to hand Goretzka a bumper deal to move to the Emirates Stadium, the Mirror report.
Schalke have attempted to extend Goretzka’s contract but he is refusing to play ball.
That could force the Bundesliga club to sanction a sale in January, as opposed to losing him for nothing next summer.
Goretzka joined Schalke from VfL Bochum in 2013.
The Germany international has scored 16 goals in 125 appearances for the club.
He has won 10 senior caps for his country.
