The deal is subject to the 38-year-old catcher passing a physical, a person familiar with the negotiations said, speaking on condition of anonymity because the agreement was not yet final. Rodriguez will receive $3 million each in 2010 and 2011.
A 14-time All-Star and the 1999 AL MVP, Rodriguez split last season between Houston and Texas. He hit a combined .249 with 10 homers and 47 RBIs.
Rodriguez, a 13-time Gold Glove winner, figures to be a backup to Jesus Flores, who missed most of last season. Flores was injured May 9 when he was hit by a foul ball, returned for three at-bats in early September, then had surgery to repair a torn labrum. He hit .301 with four homers and 15 RBIs in 93 at-bats.
Rodriguez's deal was first reported by mlb.com.
After playing for Texas from 1991-2002, Rodriguez spent one season with Florida and 4 1/2 with Detroit before finishing 2008 with the New York Yankees. He then signed with Houston, and the Astros traded him to the Rangers on Aug. 19.
Rodriguez is the major league leader in games caught with 2,288.
Earlier Monday, Washington acquired right-hander Brian Bruney from the New York Yankees for a player to be named.