The 22-year-old is a target for the Gunnres and was subject of a summer move to the Emirates Stadium which eventually fizzled out.
However, according to the aforementioned source, Mauricio Pochettino wants to beat his North London rivals to the signing of the Germany international.
Pochettino has reportedly been keeping a close eye on the midfielder, having sent a scout to watch almost all of Schalke's Bundesliga fixtures this season.
Goretzka is out of contract with Schalke at the end of the season, meaning a cut-price deal could be possible in January.
Both sides will have to move quickly though, as reports in Germany and Spain claim that both Bayern Munich and Barcelona are also keeping an eye on the youngster.