However, the team also said it would consider re-signing Jones as a nickelback and kick returner.
The 34-year-old Jones was scheduled to earn $5.5 million in 2018 in his option year. Unless he is willing to accept a cut in salary, Jones probably will become a free agent on Wednesday.
"Of course I'd love to come back," Jones told Geoff Hobson of Bengals.com. "Don't make it sound like I'm ready to get out of there, because I'm not. I'm very thankful to everyone there, coach [Marvin] Lewis and [team owner Mike] Brown.
"I'd love the chance to get on the field and compete every play, every down. Obviously I've got some things to sit down and think about the next couple of days with my family."
Jones has played 11 NFL seasons, the last eight with the Bengals.
A one-game suspension at the start of the season for violating the league's personal conduct policy and injuries limited Jones to nine games in 2017, all as a starter.
He was a first-team All-Pro selection in 2014 and was a Pro Bowl selection in 2015.
Jones has been involved in a number of on-field and off-field incidents, however, and that includes a misdemeanor charge from an altercation that led to his one-game 2017 suspension.
Jones also was suspended for the 2007 season and part of the 2008 season for violating the league's personal conduct policy.
The Bengals have several young cornerbacks who could be pushing for playing time, including Dre Kirkpatrick, William Jackson and Darqueze Dennard.