The 27-year-old Hungary international is out of contract at Championship side Bolton and will join Brendan Rodgers's side on July 1, Liverpool said on their website.
Bogdan becomes Liverpool's third signing of the close season following the arrivals of midfielder James Milner from Manchester City and Burnley striker Danny Ings.
Liverpool boss Rodgers has been looking for a goalkeeper to compete with Simon Mignolet after second choice Brad Jones was released following the expiry of his contract.
(Reporting By Michael Hann; editing by Sudipto Ganguly)
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