Drake, who turns 32 this month, was 1-1 with a 5.29 ERA in 44 appearances with the Blue Jays, Milwaukee Brewers, Cleveland Indians, Los Angeles Angels and Minnesota Twins last season.
To make room on the 40-man roster, right-hander Jaime Schultz was designated for assignment.
The Rays claimed him off waivers from the Twins on Nov. 1 and designated him for assignment less than three weeks later. The Blue Jays claimed him on waivers on Nov. 26.
In 135 games over parts of four seasons with six teams, Drake is 5-9 with a 4.59 ERA.