Laws quit Scunthorpe 12 months ago to take charge at Hillsbrough, but has had a troubled start to the season.
His job was under threat after a poor run of form, but the Owls are currently on a six-game unbeaten run. Though they are still behind Scunthorpe, who claimed all three points from their trip to Hillsbrough.
"I had a call from Burnley chairman Barry Kilby asking to speak to Brian. The approach was rejected immediately," Owls chief executive Kaven Walker said.
"We're a little bit disappointed in Burnley releasing this to the media because just over 12 months ago we spoke to Burnley asking if we could approach their manager, he said no, asked us to say nothing, and we agreed on that.
"I believed we had the same agreement. Brian's aware of their interest but why would he want to go to a smaller club anyway. We want him to stay with us "