According to the Cleveland Plain Dealer, Francona and team officials were still negotiating a contract Saturday night. The report indicated the new manager would get a long-term deal that might be for as many as four years.
Francona led the Boston Red Sox to World Series titles in 2004 and 2007. He was let go last year after a September collapse in which the Red Sox went 7-20 down the stretch to miss out on a playoff spot.
He spent this season working as an analyst for ESPN.
Francona will replace Manny Acta, who was fired in September. Sandy Alomar Jr. managed the team in the interim to finish the season. He was the only other known candidate for the job.
Acta was ousted just before finishing his third season with the Indians. He compiled a 214-266 record in Cleveland, including a 65-91 mark this year.
Francona, 53, also managed the Philadelphia Phillies (1997-2000) and was with the Red Sox for eight seasons. His overall managerial record is 1,029-915.
The Red Sox didn't get the desired results by replacing Francona, as his successor, Bobby Valentine, was fired this week after one season.
One of Francona's 10 seasons as a major league player was spent in Cleveland, as he played there in 1988. He also spent time in the Indians' front office after he was fired in Philadelphia.