Axford, from Simcoe, Ont., could replace Chris Perez, who was released on Oct. 31 following a turbulent season with the club. The 30-year-old Axford, who finished last season in St. Louis, is expected to undergo medical exams in the next few days and as long as there are no issues, he'll close for the Indians in 2014.
After releasing Perez, the Indians said they would consider using relievers Bryan Shaw or Cody Allen in that role, but decided to pursue a veteran and showed interest in Axford and Joaquin Benoit.
Axford lost his closer's job with Milwaukee after blowing six saves and was traded to the Cardinals in August. He blew seven saves last season.
The six-foot-five right-hander had 46 saves for the Brewers in 2011.