The 29-year-old was a club-record £13million signing when he joined from Norwich in January 2017, but after a bright start to his Turf Moor career he has been limited by a series of injuries, making only 21 starts since suffering a serious knee problem in December 2017.
Brady is reported to be interesting Celtic, with Southampton, Watford and Crystal Palace also said to be keen on the Republic of Ireland midfielder.
"I've really enjoyed my time here," Brady told Burnley’s website.
"I've loved every minute and I'm really going to miss the lads. They have been a massive part of my career and I've made friends who I will hold onto for a long, long time. In that respect everything has been great.
"Unfortunately, I picked up a quite significant injury when I felt like I was almost reaching the peak of my game in that season where we finished in the European spots.
"We'd had a really good spell and picked up a lot of points but then I picked up the injury and that was me for almost a year. I found it difficult with the magnitude of the injury getting back.
"It was a long road and a hard road for me, personally, but I managed to get there."
Brady has not played since suffering an Achilles problem on international duty in March, and will stay at Burnley's Barnfield training complex to work on his fitness until his next move is finalised.