The 22-year-old midfielder had been linked with a move to Villa Park, with manager Steven Gerrard reportedly present at a recent game at the Stade Velodrome.
Kamara will join the Premier League club on a free transfer at the end of his contract, which expires at the end of next month.
The Frenchman primarily plays as a defensive midfielder, but can also fill in at centre-back and was a big part of Marseille's strong 2021-2022 campaign, which saw them finish second in Ligue 1 and reach the semi-finals of the UEFA Europa Conference League.
Of midfielders in the French top division this season, only Johan Gastien and Jordan Ferri made more than Kamara's 2,383 passes, while of those who made over 1,000 passes, only five players had better accuracy than his 90.68 per cent.
After the signing was confirmed, Gerrard told Villa's official website: "I am delighted that we have been able to attract one of the most promising young talents in European football.
"We have a very clear plan to make our team stronger and Bouba is an important part of that."
The former France Under-21 international also explained that a meeting with Gerrard and other Villa representatives convinced him to join the Midlands club.
"When I met with Steven, Christian [Purslow, chief executive] and Johan [Lange, sporting director], I knew Aston Villa was for me," Kamara said.
"Their ambition and determination to succeed matches my own. I cannot wait for pre-season to get going."