Cole, however, re-signed with the Patriots on Monday. New England released rookie offensive lineman Josh Kline to create a roster spot.
Washington, signed by the team this offseason after previous tenures with the Seattle Seahawks and New York Jets, was one of five running backs to land on the team's initial 53-man roster, but depth at the position and the emergence of LeGarrette Blount as a return option may have played parts in his release.
Washington was expected to contribute as a kickoff returner this season after finishing second in the NFL in kick return yard average last season.
Cole was signed as a free agent last offseason and played in 14 games on defense and special teams, totaling 17 stops. Primarily a cornerback, he also earned reps at safety.