The team wants to get his name on something lengthier.
Tagged for the second straight year by the Browns, the dependable Dawson will return for a 14th season in Cleveland, the only steady facet of the franchise since its expansion rebirth. Dawson will make $3.8 million next season, a raise from the $3.175 million he made last season, arguably the finest of the 37-year-old's NFL career.
Dawson converted on 24 of 29 field-goal attempts, making seven kicks beyond 50 yards -- one shy of the league record. He also made all 20 extra-point tries and did so despite the Browns using three different holders and changing long snappers. Dawson had two kicks blocked and one never had a chance after the snap hit a lineman's foot.
General manager Tom Heckert recently said it's an "ultimate goal" to sign Dawson to a long-term contract. However, Dawson has heard that from past Cleveland's front offices and had promises broken as the club has endured regime changes amid more than a decade of losing.
It's not known if the Browns are in talks with Dawson's agent, Neil Cornrich, on a contract. Heckert and Cornrich did not immediately return messages seeking comment.
Dawson has said he would like to be with the Browns when they finally put it all together.
"I don't want to be Moses," he said after last season. "I don't want to lead the people right to the edge and not get to go in."
Before Dawson became a free agent after the 2010 season, he sold his home in suburban Cleveland figuring he would never be back. He was surprised when the Browns tagged him as their franchise player last March. While he played last season, his wife and three children lived in Texas.
It wasn't easy on Dawson.
"It was a tough year family-wise, but I've got a very strong wife who can handle a lot and great kids," he said one day after the Browns' season finale. "We're all in this thing together. The fun part for me is my two boys are getting old enough to where they can throw some of their input into the equation."
Dawson is the club's career leader in field goals and second (1,115 points) in scoring to Hall of Famer Lou Groza (1,349) Dawson has made 83 percent (276 of 332) of his field-goal attempts and been remarkably consistent in the tricky, windy conditions at Browns Stadium, where visiting kickers have struggled to put the ball between the uprights.