Lennie Lawrence tried to sign Coles two years ago, but now the Bluebirds are understood to have made a £200,000 offer for Coles.
However, with Championship rivals Hull City also after Coles, Bristol City are wanting more money for Coles, which Jones may not be able to meet.
Robins chairman Steve Lansdown said: "Both Cardiff and Hull have made enquiries about Danny, both of which have been rejected.
"We are not looking to sell Danny and we see his future at Ashton Gate."
Coles, however, is believed to want to leave Bristol and is not keen on the new deal offered to him after he finished the end of last season in fine form.
Coles would be an ideal replacement for the number of defenders who have left Ninian Park this summer.
Jones is desperate to add names to his squad sooner rather than later with the Bluebirds' pre-season beginning at Accrington Stanley on Saturday.
Jones is also believed to be in talks with John Curtis and Glenn Loovens (Feyernood defender) while he is still hopeful of at least a loan deal for Mikkel Bischoff from Manchester City.
Following the sale of many key players over the season Cardiff only have a first team squad of 19, which includes three goalkeepers plus largely untried players Toni Koskela, Danny Parslow, Byron Anthony and Nicky Fish.
Jones hopes to eventually work with a squad of around 23 players including two goalkeepers.
The Bluebirds face Conference club Accrington Stanley on Saturday before playing Hamilton Academicals on Monday evening, Falkirk next Wednesday and Kilmarnock on Saturday, July 23.
The first team pre-season programme will be completed by the visit of Premiership outfit West Bromwich Albion, including former Bluebirds striker Robert Earnshaw, to Ninian Park on Sunday, July 31.