Bryant, who turns 30 next month, missed only one game in the past three seasons and recorded 87 tackles, 2.5 sacks, and two interceptions over that span. He spent six years in Seattle and made 121 tackles.
He began his career as a defensive tackle but moved to end in 2010. His spot's primary responsibility is stopping the run.
Bradley spent four seasons (2009-12) as Seattle's defensive coordinator. The Super Bowl champion Seahawks drafted Bryant in the fourth round in 2008.
Though free agency doesn't begin until Tuesday, Bryant is able to sign a contract with the Jaguars because he was released by Seattle on Feb. 28. He was scheduled to make $4.5 million in 2014 and was due a $3 million bonus if he was on the Seahawks' roster on March 14.
Bryant was given a new contract during the 2012 offseason, but the emergence of Michael Bennett -- one of Seattle's top free-agent priorities -- appeared to make him expendable.
Bryant had been one of just four Seahawks players left to precede the arrival of general manager John Schneider and head coach Pete Carroll in Seattle. The only players still remaining from the pre-Carroll era are center Max Unger, defensive tackle Brandon Mebane and punter Jon Ryan.
The 6-foot-4, 323-pound Bryant was also a captain and vocal leader of Seattle's league-best defense.
The Associated Press contributed to this report.