The new deal is a timely tonic for the defender, who is fighting a battle to regain full fitness after breaking his neck in a car crash in March.
Robins chairman John Laycock revealed the news and issued a positive bulletin with regard to the player`s fitness.
Speaking to the club's official website, Laycock said: "There is good news on two fronts, firstly because Kevin has agreed to sign the contract and secondly because he is now making rapid progress with his injury.
"The metal halo he had to wear to immobilise his neck and head has been removed and he is now wearing a fixed collar instead.
"This will be replaced by a softer collar before long and all the indications are that the worst of the recovery programme is now behind him."
Jones, meanwhile, is on the brink of signing a deal with a non-league club.
The former Charlton striker just needs to negotiate a settlement with City before departing Ashton Gate.
Laycock added: "Steve doesn't want to travel far for home games and has received an offer that would suit him.
"We are close to agreement over a settlement of his contract and I expect the matter to be concluded soon."
