Franjic, who has played 39 A-League games during two seasons at City, underwent a successful medical with the South Korean top-flight club on Tuesday, according to News Corp.
While the 29-year-old right back - capped 20 times for the Socceroos - had one year remaining on his City contract, the two clubs are set to confirm the transfer agreement this week.
It leaves the Melbourne-based club with a hole to fill defensively, despite the recent signings of Iacopo La Rocca and Scott Jamieson.
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Franjic, formerly with Brisbane Roar and Torpedo Moscow, will have a fight on his hands to help newly-promoted Daegu avoid relegation from the K-League.
The club currently sits one place off the bottom, after the half-way mark of the 2017 season.
