The Bundesliga champions are among several European clubs monitoring the French youth international, who currently plays for Montpellier's under-19 side despite still being eligible for the under-17s.
Ameur has been capped by France at under-16 level and drew attention from a number of top clubs, including several Premier League sides, during this summer's International Dream Cup in Japan.
Whether the youngster ends up at Bayern's Campus remains to be seen, with interest reportedly coming from elsewhere in Europe as well as England. Bayern head coach Vincent Kompany has shown a proven eye for developing young players since taking charge, a track record that could work in the club's favour as they compete for Ameur's signature.
The teenager has been a standout performer for Montpellier's academy over the past year, regularly playing above his age group and forming part of a promising generation at the French club.
His technical quality and vision in midfield have made him one of the most closely watched prospects to emerge from the Montpellier system in recent seasons, and it is that reputation that has now put him on Bayern's radar.
Montpellier will be aware of the growing interest and face the challenge of trying to tie down one of their brightest young talents before rival clubs move to secure his signature, much as they have had to navigate with other academy graduates in recent years.