The 24-year-old Kosovo international has emerged as a possible target for Villa after RB Leipzig, his long-time suitors, cooled their interest.
Bremen are demanding a fee of around £18million for Rashica, who has scored 24 goals in 74 matches during two seasons with the German club.
It is understood negotiations with Villa, which are still at an early stage, could progress further in the coming week.
Villa boss Dean Smith is keen to add a winger before next month’s transfer deadline, having already signed right-back Matty Cash and striker Ollie Watkins.
The latter arrived in a club-record £28m move from Brentford earlier this week and could make his first appearance for the club in their pre-season friendly against Manchester United.
Villa are also edging closer to the signing of goalkeeper Emiliano Martinez from Arsenal in a deal worth around £20m, while Smith is continuing his search for another forward.
Bournemouth's Josh King and Celtic's Odsonne Edouard are among the players being considered.