Oliver, who turned 41 in October, was a key member in the Texas Rangers' bullpen the past two seasons. He was 5-5 with a 2.29 ERA in 51 innings with two saves and 16 holds in 2011. That was after he pitched 61 2/3 innings and had a 2.48 ERA in 2010 for Texas.
Oliver also provided some veteran leadership for a young pitching staff after returning to the Rangers following three seasons with Anaheim. Oliver has had a total of three stints in Texas, including his time as a starter when the club made its first playoff appearance in 1996.
Should Oliver finalize the deal with the Blue Jays, the Rangers would receive a supplemental first-round pick in next year's draft.
The Dallas Morning News first reported that Oliver was discussing a deal with the Blue Jays.
Richard Durrett covers the Rangers for ESPNDallas.com.