The two clubs have announced that they have agreed a player-exchange transfer subject to the duo passing medical and agreeing personal terms
Jones will become Cardiff manager Ole Gunnar Solskjaer's fourth signing of the January transfer window, with Nigeria forward Odemwingie heading to the Britannia Stadium in exchange.
Odemwingie only joined the Welsh club in September but has struggled to find form since his transfer from West Brom, scoring just two goals in 17 appearances.
The ex-Lille and Lokomotiv Moscow man was omitted from the Cardiff squad for the 1-0 FA Cup win at Bolton Wanderers on Saturday.
Solskjaer suggested after the game that a deal with Stoke was on the cards, and an agreement has now been reached - subject to both players passing medicals and agreeing personal terms.
A statement on Stoke's official website read: "Stoke City can confirm it has reached an agreement with Cardiff City over a move for striker Peter Odemwingie that will see Kenwyne Jones head to South Wales in a player-exchange transfer.
"The deal is subject to both players passing medicals and agreeing personal terms but the Potters hope to conclude their move for the Nigerian international over the weekend."
Trinidad and Tobago forward Jones signed for Stoke from Sunderland back in 2010 in a club-record £8 million move.
However, the 29-year-old has made just 10 appearances this term and was reportedly involved in a dispute with manager Mark Hughes prior to the 5-3 defeat to Liverpool earlier this month.