British media reported that the deal between the two Premier League clubs was worth 19 million pounds ($24.02 million).
Kellyman has made six first-team appearances for Villa. He joined the Birmingham club's youth academy from Derby County in 2022 and is an England youth international.
"It's a massive club with an amazing history, so it is great to join... I want to be successful in the Chelsea shirt, win games and hopefully lift trophies," Kellyman told the club website.
"(Over time) hopefully, I can become one of the faces on the side of the stadium. I want to become a big player for this club."
Villa signed Dutch defender Ian Maatsen from Chelsea for a reported fee of 40 million pounds on Friday. The 22-year-old was loaned out to Borussia Dortmund last season.