The Danish stopper, a key figure in the club's promotion to the Barclays Premier League in 2008-09, has lost his place in the side to Lee Grant.
Jensen, 35, only signed a new two-year contract in the summer but has grown frustrated after being left out and has now told manager Brian Laws he wants to leave in January.
A club statement said: "Burnley Football Club can confirm long-serving goalkeeper Brian Jensen, 35, has submitted a written transfer request due to a recent lack of first-team opportunities.
"The first-team management at Burnley Football Club will now consider this request."
Source: PA
