The England Under-21 cap, 20, may go out on loan in January.
St Etienne sent leading scout Romain Barq to watch him play last week while Bournemouth and Swansea have also had top spies at matches.
Balogun is in sparkling form for the Gunners' shadow side and scored for England Under-21s on November 11.
The striker joined Arsenal's academy in 2008 and quickly rose through the ranks in the club's youth level.
The 20-year-old amassed a staggering total of 38 goals and eights assists in 41 appearances for the Under-18s.
He has since registered 37 goals and nine assists in 60 matches across all competitions for the Under-23s.