Coach Pete Carroll said Avril is doing well in his recovery and is expected to start running later this week.
Asked if Avril's injury was considered serious, Carroll joked to reporters, "Well, it hurts," according to The Tacoma News-Tribune.
"You could ask him about it. The plantar fascia thing is a real uncomfortable thing. You just have to wait it out," he said, according to the newspaper.
Carroll said Avril won't be able to participate in practices for "another couple weeks at least."
Avril signed a two-year, $15 million contract with the Seahawks this offseason after playing five seasons with the Detroit Lions.
Avril had 9.5 sacks last season, and along with free-agent addition Michael Bennett was signed by the Seahawks to further boost their pass rush and also provide insurance for defensive end Chris Clemons, who is returning from a torn ACL suffered in the team's playoff victory over the Washington Redskins.
The Seahawks also will be without defensive end Bruce Irvin for the first four games of the 2013 season after he was suspended last week for violating the NFL's performance-enhancing drug policy.
Information from The Associated Press was used in this report.