West, 26, had been designated by Texas recently to clear room for the signing of Juan Carlos Oviedo. He reached the bigs for three outings last season, but spent most of the year at Triple-A. Across 56 2/3 innings, he worked to a 3.34 ERA with 10.2 K/9 and 2.9 BB/9.
The 27-year-old Burns, meanwhile, also pitched last in the majors with Texas. He worked at Triple-A last year for the Rangers and Rays, totaling 63 2/3 frames with a 4.95 ERA and 8.2 K/9 against 2.3 BB/9. Toronto claimed him off waivers in late September.