Terms of the deal were not disclosed but multiple reports said the offer sheet for the restricted free agent, who has spent his entire seven-year career with the Nashville Predators, is a 14-year pact worth about $110 million.
The Predators have one week to match the offer and keep Weber or accept compensation that would likely be four first-round draft picks.
"Our ownership has provided us with the necessary resources to build a Stanley Cup-winning team," Predators' General Manager David Poile said in a statement.
"Due to the complexity of the offer sheet, we will take the appropriate time to review and evaluate it and all of its ramifications in order to make the best decision for the Predators in both the short and long-term."
Weber, selected by the Predators in the second round of the 2003 NHL Draft, is a three-time All-Star who led all Nashville defenseman with 19 goals and 49 points last season.
The Predators have previously said that if a team entered into an offer sheet for Weber, who was also a member of Canada's gold-medal winning team at the 2010 Olympics, their intention would be to match and retain him.
The Flyers, who fell to the New Jersey Devils in the Eastern Conference semi-final this year, allowed the third-most goals against during the regular season among playoff teams.