In addition, the Jets released cornerback Lito Sheppard, a team source told ESPN NFL Insider Adam Schefter.
The move was expected, what with Sheppard's disappointing season after being acquired from Philadelphia last offseason; he also was due a $10 million roster bonus.
Edwards, acquired from Cleveland in a trade last October, is likely to remain with the Jets after being tendered at the highest possible level. He had 35 catches for 541 yards and four scores in the regular season for New York.
Washington, a Pro Bowl kick returner in 2008, is coming off a broken right leg suffered in Week 7 that sidelined him the rest of the season. He's expected to be ready for this season and could be an attractive option for a team looking for a speedy presence in the backfield.
Smith was one of the Jets' top special team players and a versatile presence on the offense, lining up at quarterback and running back along with wide receiver.
Backup quarterback Kellen Clemens received an original-round tender (second). But because the Jets used second-round tenders -- higher than their draft position -- on Washington and Smith, they would instead receive a third-rounder if another team signs him.
Original tenders were placed on safety Eric Smith (third), offensive tackle Wayne Hunter (third) and cornerback Drew Coleman (sixth). The Jets also have the right of first refusal on safety James Ihedigbo, who was undrafted.
The Jets announced this week they would release running back Thomas Jones when the free-agency period begins Friday.
Information from The Associated Press was used in this report.