Abraham was part of the Villa side that won promotion back to the Premier League in 2018/19, scoring 25 league goals while on loan from Chelsea, who sold him to AS Roma in 2021.
He was on loan at Besiktas from Roma, but the Turkish club activated their buy option on Monday for £11 million, meaning they could make a profit on the player.
The 28-year-old England international will boost Villa manager Unai Emery's attacking options as the Midlands club try to keep themselves in the Premier League title race.
Villa are third in the table, four points behind leaders Arsenal, and host Brentford on Sunday.