Pegula would not put a timetable on hiring replacements. Protected lists for the NHL Expansion Draft, to be held June 18-20, are due June 17, and the 2017 NHL Draft will be at United Center in Chicago on June 23-24.
"We're going to move as quickly and efficiently as we can," Pegula said. "Sometimes you talk to half a dozen or so people and you need to talk to more people. It's all a function of how our meetings go."
Pegula denied having already interviewed former Los Angeles Kings and San Jose Sharks GM Dean Lombardi. Lombardi was fired by the Kings on April 10.
"We haven't talked to anybody. ... It's a pure fabrication," he said.
As for what strengths the next GM should have, Pegula said he had a few ideas.
"I'll go back to my four words again: discipline, structure, communication and character," he said. "Those things are all something you need in your leaders so they are able to judge and look for the same characteristics in players and employees that work for your organization."
Bylsma was 68-73-23 in his two seasons as coach. The Sabres were 33-37-12 this season and finished 17 points out of a playoff spot. Forwards Eichel, Evander Kane, Ryan O'Reilly and Kyle Okposo missed significant time because of injuries.
Buffalo has not made the playoffs since 2011.
"We as members of the Sabres organization are all responsible for our success but accountability starts with me," said Pegula, the Sabres owner since 2011. "We are not happy with our season this year and there are no excuses. Six years ago I stood here and told Buffalo Sabres fans that the reason for existence is to win a Stanley Cup. That is still the truth. One team wins the Cup. We expected more this season. Kim [Pegula, CEO of Pegula Sports and Entertainment] and I spoke in depth with Tim and Dan, separately, and decided that our organization needed more discipline, structure, and communication in order to be successful. We will move forward with new leadership and compete in the National Hockey League for the Cup."
In addition to Murray and Bylsma, the Sabres fired director of scouting Rob Murphy and director of amateur scouting Greg Royce. Pegula said head amateur scout Jeff Crisp will oversee the amateur scouting department, and a committee of in-house scouts will head up the professional scouting department.