The 19-year-old was released by the Clarets at the end of last season following concerns over a knee injury.
However, having proved his fitness to the club, Conlan has now penned a one-year contract at Turf Moor.
"I'm very happy to be back at the club, I know a lot of the lads from the last few years and this is where I want to be at. I want to progress now with Burnley again and hopefully kick on in the next year," he told Burnley's official website.
"I had a few other offers from clubs who wanted me to go on trial but as soon as the opportunity came up here to get another contract it was what I wanted to do."
Burnley are back in Premier League action on Saturday when they make the trip to Crystal Palace.