Johnson, 31, was released by the A's last week. This is an important factor, because we shouldn't get caught up in Johnson signing a "minor-league" deal. He's making $10 million this season, paid by Oakland. The signing team of a player who has been released is only responsible for the prorated league minimum fo the remainder of the season. So the Tigers can roster Johnson as soon as they wish.
A former All-Star closer, Johnson has struggled mightily this season. He has a 7.14 ERA, 2.06 WHIP and 28 strikeouts against 23 walks in 40 1/3 innings. He was very good in 2012 and effective last season, too, so it's worth a shot to grab him on the cheap.
The Tigers rank 28th in baseball in bullpen ERA, sporting a 4.53 mark. They have 14 bullpen losses and 10 blown saves.