The source said Kansas City will continue working toward a new contract with Jones and would use the franchise tag if necessary. Teams have until March 12 to use the tag.
General manager Brett Veach said at the scouting combine this week that the Chiefs are open to using the franchise tag on Jones, but that the reigning Super Bowl champions would like to agree to a long-term deal with the Pro Bowl lineman.
"It's certainly a priority for us. Sometimes these things take some time to work through," Veach said. "The franchise tag is something that we have the ability to use. We'd certainly like to see if we can work something out with him.
"The tandem of Chris Jones and Frank Clark is something any team would want, and we're going to work hard to get that done."
Jones didn't participate in the Chiefs' offseason practices and workouts in 2019 in the hope of getting a contract extension. It didn't happen and Jones reported to training camp on time. He played the season in the final year of the contract he signed in 2016 as the Chiefs' second-round draft pick from Mississippi State.
Jones, who turns 26 in July, led the Chiefs in sacks in each of the past two seasons.
Jones failed to register a sack in Super Bowl LIV but still played a big role in the Chiefs' victory. His pressure on 49ers quarterback Jimmy Garoppolo forced an interception in the second quarter. Jones also knocked down three of Garoppolo's passes.
He has 33 sacks, 2 interceptions and 7 forced fumbles in 61 career games.
ESPN's Adam Teicher contributed to this report.