The Englishman's future is currently the subject of much speculation, with Chelsea believed to be open to a sale during this summer's transfer window for financial reasons.
Tottenham Hotspur continue to be linked with the 24-year-old, but according to The Independent, Villa are also firmly in the race for his signature during the summer market.
The report claims that Villa head coach Unai Emery is a huge admirer of Gallagher, who was one of Chelsea's most productive players during the 2023-24 campaign.
The midfielder scored seven goals and registered nine assists in 50 appearances for the Blues during the recent season, but his contract is due to expire next summer.
Villa finished fourth in the 2023-24 Premier League table to secure Champions League football for next term.
Meanwhile, according to the Daily Record, Fulham are exploring the possibility of signing Scotland international Stuart Armstrong on a free transfer this summer.
The 32-year-old's contract at Southampton is due to expire at the end of June, and he is currently being heavily linked with a move away from St Mary's.
Fiorentina are one of a number of clubs to be linked with the midfielder, but it is understood that Fulham are also looking to bring him to the Premier League.
Armstrong played a key role in Southampton securing promotion from the Championship in 2023-24, scoring five goals and registering seven assists in 42 appearances.
In total, the Scot has represented the Saints on 214 occasions, scoring 25 goals and registering 19 assists, having arrived from Celtic in June 2018.
Elsewhere, Bundesliga outfit Eintracht Frankfurt are reportedly determined to sign Brighton & Hove Albion midfielder Pascal Gross during this summer's transfer window.
The 32-year-old, who is currently with the Germany squad preparing for Euro 2024, enjoyed an impressive 2023-24 campaign, scoring five goals and registering 13 assists in 47 appearances in all competitions.
Gross is out of contract at the Amex next summer, though, and he is currently being strongly linked with a move away from the club.
According to journalist Florian Plettenberg, Frankfurt view Gross as their leading midfield target this summer, with the German outfit determined to secure a deal.
However, the midfielder is said to be focusing on this summer's European Championship.
Gross has scored 32 goals and registered 52 assists in 161 appearances for Brighton since making the move to the club from FC Ingolstadt 04 in the summer of 2017.