The Daily Mail says Wednesday have secured the services of Jones on a three-year contract.
Owls manager Carlos Carvalhal has been tracking Jones all summer and submitted a bid that matched the buyout clause in the 31-year-old's contract at Turf Moor.
Jones said: "It was a bit of a whirlwind few hours to be honest. When I arrived at training yesterday I had a message on my phone from my agent saying that Sheffield Wednesday have matched the buyout clause in my contract and that I can go for a medical in Sheffield. So I drove over to Hillsborough from Burnley and a few hours later I had completed my medical and everything had been agreed, so I was able to sign the contract.
"Wednesday is a massive club with fantastic, passionate fans and I'm delighted to be here. I can't wait to get started. I supported them growing up and loved watching the likes of Chris Waddle play."