The 37-year-old, who has made 311 appearances for the Clarets, has been at Turf Moor for a decade following his departure from West Bromwich Albion in 2003.
Manager Sean Dyche told the club's official website: "Brian has made a huge contribution to this club and I was aware, even before I came here, what a fabulous servant he has been. However, Brian understandably wants first-team football and unfortunately that is something we just could not guarantee.
"It's always sad to see long-serving players leave, but I am sure I speak on behalf of everyone connected with Burnley Football Club in wishing Brian and his family well for the future, on and off the pitch."
Jensen featured in the final minute of Burnley's 2-0 victory over Ipswich Town on the last day of the Championship season last weekend.