The Indians also will send $10 million to the Braves to help fund the $29 million in guaranteed salaries for Bourn and Swisher in 2016.
The deal cannot become official until all players clear waivers.
Johnson, 30, missed most of May with a broken left hand and is hitting .235 with two homers and 11 RBIs in 56 games. Against left-handers, though, he is hitting .323. He played in 23 games at third base and 20 games at first base for the Braves.
Johnson will make $7.5 million next season and $9 million in 2017, and his team owns a $10 million option for 2018.
Bourn, 32, is hitting .246 with 12 doubles and 13 steals this season for the Indians. Since signing a four-year, $48 million deal in 2013, he has hit just .257. Now he will return to Atlanta, where he was an All-Star in 2012.
Swisher, 34, is hitting .198 with two homers and eight RBIs in 30 games this season. He signed a four-year, $56 million deal before the 2013 season and will be paid $15 million next season.