The move likely ensures Weber will be with the team for the start of the 2011-12 season. Weber is set to become a restricted free agent July 1.
Weber made $4.5 million this past season and was nominated for the Norris Trophy, awarded to the league's best defenseman.
The Predators lost to the Vancouver Canucks in the Western Conference semifinals. It was the first time in Nashville's history the team advanced to the second round of the playoffs.
Information from ESPN.com's Pierre LeBrun was used in this report.