Axford, from Simcoe, Ont., was a combined 7-7 with a 4.45 ERA for Milwaukee and St. Louis last season, but he was 1-0 with a 1.74 ERA after the Cardinals acquired him in a late-August trade. He made six appearances in the post-season and allowed a run in 5 2-3 innings. He struck out nine.
The 30-year-old Axford led the Brewers with 46 saves in 2011 but lost the closer role a year later.
Teams had until midnight to offer 2014 contracts to unsigned players on their 40-man rosters.