The Burnley midfielder confirmed he has signed for the Owls.
Jones, who started Burnley's Premier League opener with Swansea City, in action this pre-season
The arrival of the Burnley midfielder, who hails from Southport, Merseyside, is yet to be confirmed by Wednesday, although the Mail reports he has signed a three-year deal at Hillsborough.
Carlos Carvalhal's side met the buyout clause in Jones' contract earlier this week and the 31-year-old said he can't wait to get started at his new club.
"It was a bit of a whirlwind few hours to be honest," the former Manchester United trainee told the Mail.
"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."
Waddle celebrates reaching the FA Cup final at Sheffield United's expense
Last week, Carvalhal told Sky Sports that he was keen to add another four players to his squad, taking his tally of summer signings to nine.
Among those targets was a new central midfielder, with left-back, centre-back and wide midfield the other areas the Portuguese wished to strengthen.
Owls boss Carvalhal is tasked with delivering promotion this season
As well as filling the former void, Jones, a regular for last season's champions Burnley, will add further promotion winning experience to Wednesday's squad, which reached the play-off final in May.
The former Derby County and Wolverhampton Wanderers man is one promotion shy of Kevin Phillips' record of five from the second tier.
Jones celebrates Burnley's most recent promotion to the Premier League