Financial terms of the deal are not yet known, but
a source told ESPN NFL Insider Adam Schefter that Washington signed a one-year deal.
He now joins New England, a team that finished 25th in the NFL with a 21.2 yards-per-kickoff-return average last season. Washington trailed only the Ravens' Jacoby Jones for the league lead in kickoff return yard average at 29.0 in 2012.
Originally a fourth-round pick of the New York Jets in 2006, Washington also has been used as a change-of-pace running back during his seven NFL seasons, with career marks of 473 rushes for 2,213 yards.
The team added former University of Florida running back and 2012 Olympic silver medalist Jeff Demps during training camp last season, although he spent the season on injured reserve. He was considered a top candidate to assume kickoff return duty in 2013, and may compete with Washington for that role.
Demps also has expressed a recent interest in simultaneously continuing his track career while playing for the Patriots.