Davis, 27, has spent his entire five-year career with Washington but was allowed to hit the free-agent market Tuesday.
Davis (6-foot-4, 247 pounds) would add an important piece on offense for the Browns. Coach Rob Chudzinski and coordinator Norv Turner have had great success throwing to tight ends, and Chudzinski played and coached the position.
Davis is talented — 155 catches, 1,973 yards, 12.7 average, 12 touchdowns — but carries baggage. He tore an Achilles on Oct. 21 and missed the rest of the season. He expects to be ready for training camp but it isn’t guaranteed.
He also was suspended for the final four games of 2011 for violating the league’s substance abuse policy. The Washington Post reported he failed a test coming out of the lockout, then another during the season. He had a previous failed test earlier in his career. The next failed test would bring a one-year suspension.
“The lockout made me a little bit more free, all that free time,” Davis told the Washington Post. “But still, I’ve learned. I know football’s what’s important now.”
Davis compiled a career year before the suspension, catching 59 passes for 796 yards and three touchdowns in 12 games.
He caught 24 passes for 325 yards and no touchdowns in seven games last year before the injury.
The Browns don’t plan to re-sign veterans Benjamin Watson and Alex Smith, leaving a shortage at the position.
Raw backup Jordan Cameron returns and the Browns agreed to a three-year deal with Gary Barnidge on Wednesday, but neither is established. Cameron has 26 catches in two years, Barnidge 18 in five.
According to Pro Football Focus, Davis averaged 1.96 yards per route run in 2012, second best among tight ends.
Davis is from Toledo and told 92.3-FM recently he would like a return closer to home if he isn’t back in Washington.