Boxberger is two seasons removed from being an All-Star. The 29-year-old had 41 saves in 2015, but was a candidate to be non-tendered on Friday at the 5 p.m. ET deadline to offer arbitration-eligible players contracts.
Boxberger overcame missing 80 games with a right flexor strain to post a 4-4 mark with a 3.38 ERA in 30 appearances last season with the Rays. He owns a 17-20 mark with 44 saves and a 3.19 ERA in 231 career innings.
Taylor, a right-hander, recorded a 3-4 mark with a 3.32 ERA in 13 starts for Class A Kane County in the Midwest League. The 22-year-old was a fourth-round pick of the Diamondbacks in the 2016 draft.