The goalkeeper becomes Vincent Kompany’s eighth signing of the summer transfer window.
Switzerland-born Muric joined City in 2015 from Grasshopper, with loan spells to NAC Breda,
Nottingham Forest, Girona, Willem II and Adana Demirspor during his time at the Etihad.
Muric said: “I am very happy to be here, I can’t wait to meet my new team-mates now and begin training.
“I’m 23 now, so I feel it is the right time to move on and I’m looking forward to the new season.”
Manager Vincent Kompany said: "We're delighted to welcome Arijanet to the squad, he's a player who has made great progress over the last few years and we’re hoping he can continue that here at Burnley Football Club.
"He’s a tall and strong goalkeeper with lots of qualities, that will be a nice addition to our team of goalkeepers.”
