The Coyotes had made Bryzgalov and defenseman Keith Yandle, a restricted free agent, their top priorities this offseason.
Negotiations with Bryzgalov broke down because the Russian goalie wanted to not only be a top-paid goalie in the NHL, but one of the top-paid players, according to an NHL official who requested anonymity because he wasn't authorized to release the information.
The official also said the Coyotes do not intend to sign Clackson when his contract expires on June 30.
"After several discussions with Ilya and his agent, it became very clear to us that we were not in a position to sign Ilya to a long-term contract," Coyotes general manager Don Maloney said in a statement. "We will now focus our efforts on adding another goaltender to our roster for the upcoming season."
Bryzgalov went 36-20-10 with a 2.48 goals-against average and seven shutouts, but had some shaky moments as Phoenix was swept from the playoffs by Detroit. He becomes a free agent on July 1, but the Flyers are hopeful of striking a deal with the 30-year-old goalie.
"We'll see what we can do," Flyers general manager Paul Holmgren said. "He's certainly not an old goaltender. We feel like he's got a lot of good years left in him. We'll see how it goes. When you get into a negotiation like this, you're dealing with a salary cap and you try and get a number you can live with."
The Coyotes re-signed goalie Jason LaBarbera to a two-year contract on Monday, but will likely look for a topflight goalie through free agency or a trade and leave the seven-year veteran in a backup role. He and Bryzgalov had nine combined shutouts, setting a franchise record.
The Flyers, who didn't have a single shutout last season, were repeatedly frustrated with their goalie situation in the playoffs. Sergei Bobrovsky, Brian Boucher and Michael Leighton all got a crack at winning the job, but Philadelphia switched them out as starters a whopping six times in 11 games.
Boucher won all four postseason games for the Flyers, but likely won't be back after his third stint with the organization. Leighton, who led the Flyers to the Stanley Cup finals, might also be on the way out.
Bobrovsky has been considered Philadelphia's goalie of the future and Holmgren said he wasn't sure if that would change if the team signs Bryzgalov.
"We'll see if we can get a deal done with Ilya," he said. "I still think the world of Sergei. I think he's one of the top young goalies in the game. We'll continue to try and make our team better. With the position we put ourselves in with Ilya, if we can get him signed it will make our team better."
AP Sports Writer Dan Gelston in Philadelphia contributed to this report.