Danish veteran Jensen has signed a 12-month contract extension with the club.
“We always wanted him as part of the squad and his experience will be vital,” said Burnley boss Eddie Howe.
“He has a big part to play in being a role model for the other players and the young group of goalkeepers we are bringing through.
“It’s important to have good professionals around them.
“We were also delighted to offer him a testimonial, which is recognition of the outstanding service he has given to the club.”
Burnley are considering letting Stewart leave on loan again to continue his development.
“My plan is not to hang around,” said Stewart. “I am going to push myself to try and get in this team or get a club on loan.
“That’s what’s going to happen. I don’t want to waste any more time. I’m 23 now so I want to push on.”
