Saints tight end Jimmy Graham tweeted: "I want to see the evidence and hear an explanation."
Reggie Bush, a running back who played for the Saints from 2006-10 and now is with the Miami Dolphins, wrote on Twitter that the suspensions were "outrageous" and "Next thing you know we'll be playing two hand touch football! (hash)Lame"
In a memo sent Wednesday to the NFL's 32 teams, Goodell reminded them that "any program of non-contract bonuses, however it is characterized, is a violation of league rules" and said that every head coach must review those rules with assistants and players during mini-camp or preseason training camp.
Also, all players will be told how they can confidentially report rules violations.
In March, Goodell made Payton the first head coach suspended by the league for any reason, for trying to cover up the system of extra cash payouts. Goodell also indefinitely banned Williams, who was hired in January to run the St. Louis Rams' defense. Loomis was barred for eight games; Vitt for six. The Saints were fined $500,000 and lost two second-round draft picks.
Fujita, Hargrove and Smith are allowed to participate in offseason activity, including preseason games, before their suspensions take effect. Vilma, though, is suspended immediately and will be reinstated after the coming season's Super Bowl is played in his team's city.
"Nothing can be gained from sharing how I feel about" his teammates' penalties, Saints right tackle Zach Strief said. "I will miss Jonathan very much. Knowing him personally, he's a good person. This is going to be a tough thing for him to go through. In terms of his leadership, somebody else will step up and take over."
AP Sports Writers Rachel Cohen, Brett Martel, Chris Jenkins, Tom Withers and Jon Krawczynski contributed to this report.
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