A source told ESPN that Swisher agreed to a four-year, $56 million contract that also includes a $14 million option for a fifth season that will "easily vest, making the potential total value of the contract $70 million.
"Wow! What a crazy few weeks. Hey Cleveland! Are you ready? Because I'm coming home! #RollTribe," Swisher tweeted Sunday morning.
The Indians will lose their second-round pick in next year's amateur draft as compensation for signing Swisher, and the Yankees will get an extra selection following the first round.
Swisher's contract agreement was earlier reported by the New York Daily News.
The 32-year-old Swisher spent the past four seasons with the New York Yankees.
He thanked Yankees fans for their support in a follow-up tweet Sunday.
He wrote: "Thank you NY for the kindness & support you've shown me the past 4 years. It's been an amazing experience & I've loved every minute of it!"
Swisher immediately fills a hole in right field for Cleveland, which traded Shin-Soo Choo to Cincinnati this month and has been trying to improve a team that collapsed in August and finished 68-94.
A switch-hitter, Swisher batted .272 with 24 homers and 93 RBIs in 148 games last season, his fourth with the Yankees.
The Indians have been in the market for an outfielder throughout the offseason. They offered Shane Victorino a four-year contract worth a reported $44 million before he signed a three-year, $39 million deal with Boston.
It's been a busy offseason for Indians general manager Chris Antonetti, who is trying to fix a team that has lost at least 93 games in three of the past four seasons.
Antonetti fired manager Manny Acta and replaced him with Terry Francona, the former Boston manager who has spent the past few weeks meeting with his new players. Antonetti also signed free agent first baseman Mark Reynolds and was part of a three-team, nine-player deal that sent Choo to the Reds for Stubbs and brought Cleveland prized pitching prospect Trevor Bauer from Arizona.
Swisher and his wife, actress JoAnna Garcia, who is expecting the couple's first child in May, were at Progressive Field on Dec. 18 for a tour of the team's facilities, including the club's family room and a lunch with Francona, team executives and former Ohio State football coach Jim Tressel.
Swisher and his wife were shown a suite and the team's clubhouse before the Indians tugged at Swisher's Ohio State connections to convince him Cleveland should be his next baseball home. Swisher played baseball for the Buckeyes.
The Indians played a video on their scoreboard that featured current Ohio State football coach Urban Meyer and basketball coach Thad Matta, who encouraged Swisher to join the Indians. Swisher was moved by the gesture and didn't expect to see Tressel at lunch in the stadium's Terrace Club.
Information from The Associated Press was used in this report.