The 6-foot-5 right-hander was selected by the Cubs in the 35th round of the 2011 First-Year Player Draft out of Lafayette College (Pa.).
The Nationals optioned Rodriguez, who had a powerful arm but often lacked control, on June 4. Rodriguez, 26, spent three seasons in Washington and had a 0-1 record with a 4.00 ERA in 17 appearances this year.
Ian McCue is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.