Sean Payton's season-long suspension for New Orleans' bounty system begins at the end of the week.
NFL Commissioner Roger Goodell said Monday an appeal by Payton could leave him in charge of the Saints -- at least temporarily.
"I said in a letter they have to appeal by April 2, I believe," Goodell said at the NFL Annual Meeting 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."
Otherwise, Payton will be gone until after the Super Bowl for his involvement in the bounty program that ran for three seasons, including the team's 2009 championship run.
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."
Payton's agent, Donald Yee, said "no decisions have been made about an appeal" by his client.
Goodell also is waiting for recommendations from the players' union before punishing any players for participating in the bounty system that targeted opponents for big hits. He has discussed the bounties with NFLPA leader DeMaurice Smith.
"Yes, they can't take somebody from another staff," Goodell said. "But they could take somebody who's not employed right now in the NFL."
Fox Sports insider and NFL Network contributor Jay Glazer reported Monday that Payton and Parcells planned to meet this week to discuss the possibility of Parcells leading the Saints for the 2012 season. However, Parcells said Monday he doesn't believe he "has the bug" to coach again.
If Parcells, a 2012 Pro Football Hall of Fame finalist, returns to coaching in any capacity, he would have to wait another five years to be considered for Hall of Fame induction, Hall of Fame vice president of communications Joe Horrigan confirmed to NFL.com.
Goodell also confirmed the team would have to comply with the Rooney Rule, which requires teams to interview minority candidates, before making a interim coach hire.
The Associated Press contributed to this report.