The 21-year-old has scored 18 goals and registered four assists in 46 appearances for his current side, including 16 goals and three assists in 35 outings during the 2023-24 campaign.
Beier's future has been the subject of much speculation in recent weeks, with a number of clubs said to be keen to secure his signature during the summer market.
According to BILD, Villa are in a strong position to sign the forward this summer despite the likes of Liverpool and Manchester United being present in the race.
The report claims that Hoffenheim need to sell Beier for financial reasons, and a departure could potentially occur before the end of the month.
Villa are preparing to play in the Champions League during the 2024-25 campaign courtesy of their fourth-placed finish in last season's Premier League.
Bayern 'keen' on summer deal for Mamardashvili
Meanwhile, according to Fichajes, Valencia goalkeeper Giorgi Mamardashvili is wanted by Bayern Munich during this summer's transfer window.
The 23-year-old caught the eye with his impressive performances for Georgia at Euro 2024, and Valencia are believed to be open to selling him this summer.
The report claims that Bayern are keen on the goalkeeper, with Los Che thought to be willing to let him depart for a fee in the region of €35m (£29.5m).
Mamardashvili made the move to Valencia in the summer of 2021, and he has represented the Spanish outfit on 100 occasions in all competitions, keeping 31 clean sheets in the process.
Bayern are believed to have first expressed an interest in signing the stopper last summer.
Como 'open' talks over Moreno move
Elsewhere, Como have allegedly opened discussions with Alberto Moreno over a move for the free agent.
The 32-year-old's contract with Villarreal expired at the end of June, and a number of clubs are said to have held talks with him over a switch this summer.
According to Fabrizio Romano, Como are now firmly in the race, with the Italian outfit holding positive discussions with the left-back over a move to Stadio Giuseppe Sinigaglia.
Como will operate in Serie A next season, having earned promotion from Serie B last term.
Moreno made the move to Villarreal from Liverpool in 2019, and he represented the Yellow Submarine on 124 occasions in all competitions, including 33 outings during the 2023-24 campaign.
Como are also in discussions with Raphael Varane over a move for the ex-Manchester United centre-back.