Washington, 32, played last season for the Houston Texans, who are coached by former New England offensive coordinator Bill O'Brien and run an offensive system similar to the Patriots.
The Patriots hosted Washington on a free-agent visit March 14, as they looked to add another pass-catching target after signing Chris Hogan to a three-year, $12 million contract three days earlier.
A 12-year veteran, Washington also spent six seasons with the Tennessee Titans and four seasons with the Pittsburgh Steelers. The Steelers originally signed Washington as an undrafted free agent out of Tiffin University in 2005.
With the Texans, Washington quickly picked up the offense and became the team's starter opposite star receiver DeAndre Hopkins.
Washington caught 47 passes on 95 targets last season for 658 yards after signing a one-year deal with Houston. Washington played in 14 games, missing two because of an injury.
ESPN Texans reporter Tania Ganguli contributed to this report.