The Iceland international - who is currently representing his nation at the World Cup - is considered to be a squad player by manager Steve Bruce, with just 11 starts and 14 substitute appearances coming in the Championship last season.
However, the club's financial problems have left the future of the club's first-team squad in doubt and it appears that Bjarnason is already attracting attention from elsewhere.
According to Tuttosport, Parma want to hand Bjarnason a return to Italy, with the 30-year-old having already spent time at Pescara and Sampdoria.
Bjarnason still has two years remaining on his contract, but Villa may take the chance to cash in on the player should they receive an acceptable offer in the coming weeks.
Parma have just secured a return to Serie A for the first time since 2015.