Chamberlain, a former teammate of Alex Gordon's at Nebraska, was let go by Toronto on Friday. He had been pitching for Toronto's Triple-A affiliate in Buffalo.
Rodriguez was designated for assignment by the Rangers late last month. He had a 4.90 ERA for Texas this season.
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.