The Bengals' offer Tuesday reportedly was for $1.37 million. Since Dillon is a restricted free agent, the Bengals have the right to match any offer he receives from another team.
If the Bengals choose not to match another offer, the other team would have to give Cincinnati first- and third-round picks as compensation.
"We plan on negotiating with him and agreeing to a long-term contract. That's our intention, but that's something that will take place later," general manager Mike Brown said.
After missing the final game of the season with a knee injury, an angry Dillon said he was done playing in Cincinnati. He later softened his stance.
"We would like to have him back," Brown said.
There have been reports that the Cleveland Browns might try to sign Dillon.
The Bengals also offered contracts to their other restricted free agents: tight end Steve Bush, linebackers Canute Curtis and Billy Granville, cornerback Ty Howard, defensive tackle Andre Purvis, guard Brock Gutierrez and running back Sedrick Shaw.
Dillon ran for 1,200 yards and five touchdowns on 263 carries last season, earning his first Pro Bowl berth. He became the 11th running back in NFL history to rush for 1,000 yards in each of his first three seasons.
In just three years, Dillon is fourth on the Bengals' career rushing list with 3,459 yards and needs 269 yards to surpass Harold Green. Prior to Dillon's arrival in 1997, Green had been Cincinnati's last 1,000-yard rusher with 1,170 yards in 1992.
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