The 33-year-old has been training with Sean Dyche's side for the last week, having been released by Preston North End over the summer, and has now put pen to paper on a short-term deal.
"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 the Burnley website. "It seems like an honest place and a very hard-working place and I appreciate being a part of it.
"I have seen the first two games Nick has played and he's done very well. I have a lot of respect for that. I am here to try and sharpen up the competition in the goalkeepers' department and we'll see what happens down the line.
"I feel good physically and I still want to play. I come here with a good spirit and I have a lot of enthusiasm for getting started. I sensed from the first day I came in there was a good atmosphere, that things are in a good place, and it has paid off with a good start to the season."
Lindegaard made 29 appearances in his five seasons with United before leaving for West Bromwich Albion in 2015.