Begovic signed for Chelsea from Stoke City in 2015 and went on to make 33 appearances for the club. He began his career at Portsmouth and spent time on loan at Bournemouth in 2007.
"AFC Bournemouth is an exciting, ambitious club which has everything in place for a very bright future," Begovic said in a statement on Bournemouth's website (www.afcb.co.uk).
"Being part of a title-winning team at Chelsea was amazing but I felt now was the right time to move on because I want to be playing regular football."
Bournemouth ended the Premier League season in ninth place, securing top-flight football for a third successive term.
"Asmir is an outstanding goalkeeper who has proved himself capable of performing at the highest level over a number of years," Bournemouth manager Eddie Howe said.
"He is at the right age and the right fit. He is a leader as well and exactly the kind of character you want in the team."
(Reporting by Simon Jennings in Bengaluru; editing by Ken Ferris)
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