The 33-year-old, who left Preston at the end of last season, has been training with the Clarets following Tom Heaton's shoulder dislocation and has now agreed a deal until the end of the season.
Lindegaard spent five years at Old Trafford without establishing himself in the first team and will now challenge Nick Pope and Adam Legzdins in Heaton's absence.
“I’ve had a good week here and I’ve had a very good impression of the whole club and all the people here. It’s not hard to fit in nicely here,” Lindegaard told Clarets Player HD.
“It seems like an honest place and a very hard-working place and I appreciate being a part of it.