Oliver spent the last two years in Texas, going 6-7 with a 2.40 ERA with three saves in 125 games over the last two seasons as the Rangers advanced to the World Series in back-to-back years.
Oliver appeared in 61 games for the Rangers in 2011, plus another eight postseason games. He started the 10th inning of Game 6 of the World Series, giving up back-to-back singles with the Rangers ahead by two runs and three outs away from the team's first World Series title. He exited after Kyle Lohse's sacrifice bunt put the tying run in scoring position. That run would eventually score in the team's 10-9 loss in 11 innings.
The Blue Jays currently have just two left-handed relievers on their roster, Evan Crawford and Luis Perez. Perez, a rookie in 2011, appeared in 27 games with a 5.12 ERA. Crawford, 23, hasn't pitched above Double-A and was placed on the 40-man roster to protect him from the Rule V Draft.For more baseball news, rumors and analysis, follow @EyeOnBaseball on Twitter, subscribe to the RSS feed and "like" us on Facebook.