The 22-year-old joined Spain’s Alavés in 2017, but has spent most of his time out on loan since then.
He has spent the 2019-20 season at St. Gallen following spells with Sochaux and Almeria.
His performances in Switzerland, where he has scored nine goals in 16 games, have caught the eye of many clubs.
Having narrowly avoided relegation, Saint-Etienne are looking to strengthen and have Demirovic in their sights.
They want him to occupy the number nine position, believing he can carry his form from Switzerland to Ligue 1.
Although he is due back at Alavés this summer, St. Gallen manager Peter Ziedler believes Demirovic’s future is elsewhere.
“He still belongs to Alavés,” Ziedler told L’Est Republican. “He is sought by many clubs in the Bundesliga… and Saint-Etienne, among others, in France.”
German-born Demirovic has represented Bosnia at youth level, scoring four goals in 12 appearances for the under-21s.