Klopp succeeded Rodgers as Liverpool manager in October 2015, and has turned the Reds into arguably the world's most feared football team.
The seven-bedroom house in Formby in Merseyside was put up for sale by Rodgers last summer, says the Daily Mail.
Klopp has been living in the same house where Rodgers was based when he was Liverpool boss, with the German renting the property from his predecessor.
And the club have now bought the property for £3.75m.
Liverpool are believed to be allowing Klopp to live rent free in the house, although he will pay tax on its rental value as the perk is regarded by HMRC as an addition to salary as a benefit-in-kind.