The 26-year-old was 3-0 with a 3.00 ERA in 21 innings with the Yankees this past season, and he appeared in another 19 games and logged 26 innings in the Minor Leagues. New York designated the right-hander for assignment on Friday.
In 2013, Claiborne saw considerable big league action, appearing in 44 games and tossing 50 1/3 innings for the Yankees.
Miami is seeking as much relief depth as possible during an offseason in which Hatcher was dealt to the Dodgers as part of the Dee Gordon trade, and Anthony DeSclafani was traded to the Reds for Mat Latos.
Claiborne is arbitration-eligible in 2017.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro. This story was not subject to the approval of Major League Baseball or its clubs.