Damon told ESPN Insider Jim Bowden on Thursday that a deal has been finalized.
Damon gets a pro-rated $1.25 million in base salary with $1.4 million in performances bonuses, a source familiar with the negotiations told ESPN The Magazine senior writer Buster Olney.
Agent Scott Boras has been in talks with Indians general manager Chris Antonetti, who has been looking for a player to help a Cleveland team off to a 1-4 start and batting just .176 -- worst in the majors. The deal could include an opt-out clause once Indians center fielder Grady Sizemore returns from the disabled list.
Sizemore, who underwent minor back surgery last month, is not eligible to come off the 60-day disabled list until June 3.
The 38-year-old Damon, who is 277 hits shy of 3,000, could provide a stopgap in center or if he produces, the Indians may want to keep him for the entire season.
Damon, who could play left or center field, must pass a physical. And although he's in excellent shape, he will likely need some time to get ready to play after missing spring training.
Damon played with Tampa Bay -- his sixth AL team -- last season. He batted .261 with 16 homers and 73 RBIs in 150 games, 135 as the club's designated hitter. If Damon signs with Cleveland, the Indians would be his fourth team in four years.
The Indians need someone to ignite their lineup.
Cleveland went 1-for-11 with runners in scoring position during a 10-6 loss to the Chicago White Sox on Wednesday. The Indians set a season-high for runs and hits but didn't get any in the clutch.
Afterward, manager Manny Acta said he wasn't concerned about the slow start.
"Five games is not going to make me panic about my bullpen, my offense, defense or anything like that," Acta said. "We have to give it a little more time."
The Indians traveled to Kansas City on Thursday night. They begin a three-game series against the Royals on Friday.
A career .286 hitter, Damon has also played for Kansas City, Oakland, Boston, the New York Yankees and Detroit. One of baseball's good guys, Damon's presence alone in Cleveland's clubhouse will benefit the young Indians, who can learn from his work ethic seek advice as they did last season when slugger Jim Thome returned to the Indians.
He will start with a minor league contract and get the big league deal when he is added to the 40-man roster.
In addition to his base salary, Damon would get $100,000 each for 100 plate appearances, 200 plate appearances and each additional 25 up until 500.
Information from ESPN Insider Jim Bowden, ESPN The Magazine senior writer Buster Olney and The Associated Press was used in this report.