The promotion-hopefuls are preparing for this season’s Championship play-offs as they bid to reach the Premier League and have already identified a target to strengthen their attack.
Cardiff City, who are already promoted to the top flight for next season, are also said to considering a move for the 25-year-old.
Füllkrug has impressed for Hannover in the Bundesliga this season, boasting 13 league goals for Die Roten who face a battle to hold onto their star man in the summer.
Which side he joins may depend on who seals promotion. Fulham and Villa could in the playoffs final if they beat Derby County and Middlesbrough respectively.
Born and raised in Hannover, the German u20 international joined the Bundesliga outfit from FC Nuremburg in 2016 and has gone on to make 64 appearances, contributing 26 goals in total.
The centre-forward is rated at £4.4m, so he wouldn’t break the bank, but it’s unknown which club he prefers from the many suitors being linked with his signature.
Fulham will lose Aleksandar Mitrovic this summer when he returns to parent club Newcastle United, so they might be able to promise Füllkrug first-team football in the Premier League.
Aston Villa have stiff competition, however, with Lewis Grabban, Jonathan Kodjia, Scott Hogan and Keinan Davis in the attack, so game time might be harder to come by.