The long-serving keeper has withdrawn his transfer request following talks with new boss Eddie Howe.
“We have had a good chat with Brian and talked about a lot of things,” confirmed Howe.
“Brian is frustrated because he wants to play but his attitude is that he is going to get his head down, really knuckle down in training and try and earn his place back.
“He is focused now on Burnley and, for the short term at least, Brian will stay.”
Jensen asked to leave in December after losing the number-one slot to Lee Grant.
Grant has since cemented his place in the side - keeping two clean sheets in the Clarets’ current unbeaten four-match run - but will now continue to face competition from the big Dane.
“You want everybody singing from the same sheet. Brian is a great pro, a very good goalkeeper and we are delighted to have him with us. We just need him and Granty to fight for that number one shirt,” Howe added.
Meanwhile, the Clarets’ boss revealed that if any major transfer business is done before the end of the month it will be to add to his squad, with a couple of potential targets in his sights.
However, Howe declined to comment on on-going speculation concerning a move for highly regarded Swindon Town striker Charlie Austin.
He said: “Hopefully there will be none out. We wouldn’t want to lose anybody. As for incoming, we are talking to a couple of parties but, as with all these things, they will probably end up being late decisions.
“We are working in it and we will wait and see what happens.
“With the squad and how well they have done in the two games there is no urgent panic.”