Alexander, who turns 30 on March 1, has a strained right rotator cuff. He went 2-3 with a 5.40 ERA last season while making 18 appearances, including 11 starts.
Wilson, 35, went 0-1 with a 2.45 ERA in five April relief appearances for the Cincinnati Reds last season. He underwent Tommy John surgery in June.
Wilson owns a career record of 33-24 with a 3.41 ERA and 18 saves in 527 relief appearances over 11 seasons.