The experienced shot-stopper was released by the Clarets earlier this week, but already appears to have found his next club, with a move to Munich seemingly close.
Speaking to Bild, Kiraly said: "The club  has huge potential and great fans, as well as the prospect of reaching the Bundesliga.
"I have played or trained every day for the last 16 years. That's why I wanted a new challenge this year.
"I played for Hertha Berlin in the Champions League and in the Premier League in England and played like I could climb to the peak of a mountain.
"I'm at full fitness now and, with 1860, I hope to go to the top of the mountain again."
Kiraly had enjoyed a long spell in England, also playing for Crystal Palace in the top tier of English football.
The 33-year-old spent time on loan at Bayer Leverkusen last season.