Jones celebrates promotion to the Premier League last season
Jones missed just five matches as the Clarets secured the Championship title last season and started Saturday's Premier League opener against Swansea City.
Three days later, however, the midfielder was back in the second tier with Wednesday, who met the buyout clause in his contract.
That deal was due to expire next summer and while Dyche insisted Jones would have been offered a new one, they were unable to match the terms offered by the Owls.
"His contract was running out. We would have offered him a stronger contract and a longer contract but it wouldn’t have been three years,” the 45-year-old told the Lancashire Telegraph.
Dyche before the game against Swansea
"It’s not club policy but it’s just the way I would have seen it. I explained that to him.
"He’s a fantastic fella. He’s a man, he acts older than his age, he’s very mature and very professional, and he totally understood the situation. He felt it was right for him to make the jump now."
Jones could make his debut for Wednesday, the club he grew up supporting from Southport, Merseyside, in Saturday's Yorkshire derby at home to Leeds United.
The 31-year-old, who has been promoted to the Premier League on four occasions, is Carlos Carvalhal's sixth summer signing.
Carvalhal has been busy again this summer
And the Portuguese is set to make it seven, telling BBC Radio Sheffield that the capture of a new centre-back is imminent.
Frenchman Vincent Sasso has been heavily linked with a return to Hillsborough, having spent last season on loan there.
(Right to left) Glenn Loovens, Fernando Forestieri, Sasso and Atdhe Nuhiu react to last season's play-off final defeat