The 26-year-old Croatian international appeared in 54 matches for Atletico over two seasons with the La Liga club, reports EFE news agency.
Media outlets said on Tuesday that Atletico will receive 7 million euros ($8 million) up front and a total of 25 million euros ($29 million) if Inter decides to exercise the purchase option next spring.
Vrsaljko was part of the Croatia squad that finished second in the 2018 World Cup, losing the final to France.
To replace Vrsaljko, the Atletico picked up Colombian defender Santiago Arias from PSV Eindhoven.
Arias, also 26, signed a five-year contract, Atletico said on Tuesday.
The Colombian, who scored nine goals in 136 matches with PSV, was voted player of the year in the Dutch first division last season.
Arias started all four of Colombia's contests in the 2018 World Cup.
Atletico has been active in the summer transfer window, bringing onboard Rodrigo Hernandez, Antonio Adan, Thomas Lemar, Nehuen Perez, Gelson Martins and Arias.
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