The Lambs were relegated to from the National League North following a poor second half of the season.
They will play Southern League football next term after failing to win their battle against the drop.
Skipper Jones is one of a number of players told they can go and the former Mansfield Town man told the Tamworth Herald it was the club’s decision.
“I had my meeting with the club and have basically been told there will be no deal for me,” said Jones to the website.
“With relegation, I think we were all expecting things to happen in terms of the squad.
“Ultimately, the club has to make decisions now that are in the best interests of them moving forward.
“From my point of view, I have enjoyed my time at Tamworth and wish everyone there well.”
He added: “The biggest disappointment was the way it went for us at the end of the campaign.
“There was clear ability and talent within the squad but it never felt like we had the right blend of players.”