The Potters have confirmed the deal is subject to both strikers passing medicals and agreeing personal terms, but they hope to conclude their move for Odemwingie over the weekend.
The Nigeria international only joined Cardiff in September, moving from West Brom for a fee believed to be in excess of £2m, and scored twice in his first four games only to then fail to add to that tally in 13 appearances since.
Jones was fined and dropped earlier this month after sending a text message to Mark Hughes the night before Stoke's 5-3 home league defeat to Liverpool on January 12 saying to say he was not available to play.
Speaking after his side's 1-0 FA Cup fourth-round victory at Bolton, Cardiff boss Ole Gunnar Solskjaer said: "There is potentially a deal to be done, with Peter going to Stoke and Kenwyne coming to us.
"Obviously, you cannot confirm anything until papers are signed and medicals are done, but we are quite a long way down that road.
"I can't say anything about any other players, but I know Stoke have brought out a statement so I can confirm what they have said."