The Indians will get minor league outfielder Alex Call in the deal, which is expected to be finalized Saturday, reports said.
Alonso, 31, hit .250 with 23 home runs and 83 RBIs last season for the Indians. A nine-year veteran, Alonso was an All-Star in 2017, when he hit 28 home runs and had 67 RBIs for the Oakland A’s and the Mariners.
Chicago already has Jose Abreu at first base, but he played in just 128 games last season because of injuries and could see more time at DH with the addition of Alonso.
The White Sox also have expressed interest this offseason in free-agent slugger Manny Machado. Alonso is the brother-in-law of Machado.
For the Indians, the trade moves Alonso’s $8 million salary for 2019 off their payroll, giving Cleveland more flexibility if it wants to keep its deep rotation intact.
Alonso’s contract includes a 2020 club option at $9 million with a $1 million buyout. The option becomes guaranteed if Alonso has 550 plate appearances next year or 1,100 in 2018 and ’19 combined, and he passes a physical at the end of next season.
Call, 24, was a third-round pick of the White Sox in 2016. He was promoted to Double-A Birmingham in June and hit .242 with 12 homers and 58 RBIs in 123 games.
News of the Alonso trade was first reported Friday by Stadium.
The Associated Press contributed to this report.