"I said in a letter they have to appeal by April 2, I believe," Goodell said Monday at the owners meetings in Palm Beach, Fla. "If he decides to appeal, I probably will allow him to continue, and I would expedite the hearing and I would expedite my decision.
"We did meet twice and went through the information. If he has something else for me to consider, I will."
Payton's agent, Donald Yee, said "no decisions have been made about an appeal" by his client.
"Sean fully supports the league's player safety goals," Yee said. "Given this, he probably won't address the entire league" when he arrives at the owners meetings.
Saints spokesman Greg Bensel said Payton was expected to speak later this week at the meetings. NFC coaches are scheduled to meet the media Wednesday morning. Payton has not addressed the media since the suspension but has issued two written statements apologizing for the bounty system.
Goodell also is waiting for recommendations from the players' union before punishing any players for participating in the three-year bounty system that targeted opponents for big hits. He has discussed the bounties with NFLPA leader DeMaurice Smith.
Goodell said the league approached the Saints before their playoff game with the Lions in January and warned them to make sure the bounties had ceased. He sent Jeff Pash, the NFL's lead counsel, and Jeffrey Miller, its director of security, to speak with owner Tom Benson.
"The point was to make it clear we had new and credible info, and he should make it extremely clear with the game the next day there should be no bounty system in place," Goodell said.
Asked if Payton's punishment was as much for lying to him as it was for the actual bounties program, Goodell told a packed news conference about a pattern of untruths.
"This is a violation of a very serious rule," Goodell said. "We have made player health and safety very clear as a priority. During the process of when this first was raised two years ago, there were denials. They were not forthright and that continued through our investigation.
"This is something with zero tolerance and is not acceptable."
The league's investigation is continuing, Goodell said.
"We will continue to follow on any information we have," he said, "and will look for violations. If we get info, we will follow on it.
"Let's be clear. We discussed it with the clubs today, and I think it is fair to say non-contract bonus payments were happening around the league … and that'll discontinue."
Goodell said the Saints are free to go outside the organization to hire a coach. Payton's assistant head coach is Joe Vitt, who was given a six-game suspension, while general manager Mickey Loomis received eight games. The team was fined $500,000 and lost second-round draft picks this year and next.
Gregg Williams, who as Saints defensive coordinator oversaw the bounty pool, has been suspended indefinitely. He left New Orleans for the same coaching position in St. Louis earlier this year.
Bill Parcells said Monday he has not been approached about becoming interim coach of the Saints, but he has spoken to Payton about how to handle the season. The coach who led the New York Giants to two Super Bowl victories is a mentor to Payton and hired Payton as an offensive assistant in Dallas in 2003.
Goodell termed "inaccurate" reports that one player turned in the Saints in the bounties investigation.
"You're assuming it's a player," he said. "We have several sources on this. We're not disclosing who are sources are."
Meanwhile, new lines of black-and-gold T-shirts that read "Free Sean Payton" and "Free Payton" have hit stores in New Orelans.
Players, too, have expressed support for their embattled coach. Starting right tackle Zach Strief has said the Saints owe it to Payton to live up to the standards he has set and win in his honor.
"This season is going to take on an us-versus-the-world mentality," Strief said last week. "This team is going to come together closer than it has ever been, and I think that we all feel like we have something to prove. We're going to do our best to make this situation into a positive one."