The person spoke on condition of anonymity because there was no official announcement. The Tigers played at Yankee Stadium on Tuesday night.
Johnson had a major league-best 51 saves for Baltimore in 2012, and then tied for the big league lead last year with 50. The Orioles traded him to Oakland last winter, and he wasn't able to regain his form with the Athletics.
The 31-year-old right-hander was 4-2 with a 7.14 ERA in 38 games with the A's. Johnson was designated for assignment on July 24 and released the following week.
Johnson is likely to pitch a few games at Triple-A Toledo, hoping to join the AL Central-leading Tigers.