Steven Lawless has been given a new deal by Partick, a few days after being handed a ban for breaching betting rules.
Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts.
Jankovic, 29, becomes the latest experienced addition to a newly-promoted Gialloblu squad following the arrivals of veteran striker Luca Toni and former Palermo midfielder Massimo Donati.
A statement published on Hellas' official website read: "Hellas Verona have acquired the registration of Bosko Jankovic from Genoa CFC.
"The player has joined his team-mates at the summer training camp in Val Ridanna. After meeting (coach) Andrea Mandorlini and all the staff, he joined in with a training session for about an hour."
The club did not reveal the length of the contract Jankovic had signed at the Stadio Marc'Antonio Bentegodi.
The Serbian, who made 84 league appearances in five years with Genoa and captained the Ligurian side in the 2012-13 season, has represented his country 31 times, scoring five goals.