The deal could be completed by the end of Thursday pending a review of Soriano's medical records by the Rays, a source told ESPN.com's Jerry Crasnick.
Soriano underwent Tommy John surgery in 2004 and spent time on the disabled list with elbow tendinitis in 2008.
Soriano, who had 27 saves and a 2.97 ERA for the Braves last season, had surprised the team by accepting salary arbitration. His arrival would bolster the Rays' bullpen.
Chavez was 1-4 with a 4.01 ERA in 73 appearances last season.
Information from The Associated Press was used in this report.