The 20-year-old from Savannah, Georgia, made his Aston Villa debut against Fulham in the FA Cup on Jan. 4, 2020, and made his Premier League debut that Jan. 18 at Brighton.
"Indiana is an exciting young talent who has a lot of potential," Miami sporting director Chris Henderson said in a club statement. "He's comfortable and confident on the ball and fluid with his movement. We feel this is the right place for Indiana to further his development and reach his potential."
He made four Premier League appearances plus one each in the FA Cup and League Cup during the 2019-20 season, then was loaned to third-tier Burton for the first half of the 2020-21 season and had no goals in 12 league appearances.
He spent the second half of the season on loan to fourth-tier Cheltenham and also had no goals in 12 appearances.
Vassilev played for the U.S. at the 2017 Under-17 World Cup.