The 26-year-old faces an uncertain future at the Etihad Stadium, having struggled for regular game time under Pep Guardiola so far this season, with the Catalan club understood to be monitoring his situation.
Despite that, Barca are looking to bolster their squad going forward and Sterling is believed to be at the top of their wishlist.
The Englishman's potential arrival at Camp Nou would likely depend on a departure, and Dembele is seemingly the first in line to head out of the exit door.
Dembele, 24, will see his contract expire next summer and it is thought that Man City would consider his arrival in any potential deal involving Sterling moving in the opposite direction.
Sterling, who has less than two years remaining on his current deal, has made 302 appearances across all competitions for the Citizens, scoring 115 goals and providing 88 assists.