Giants running back Saquon Barkley was offered an extension worth "around $12M/season during the bye week" according to Ralph Vacchiano of Fox Sports.
Vacchiano adds that he "continues to hear" that Barkley wants near Christian McCaffrey's $16M average salary and that the franchise tag may be realistic for him. The tag for running backs is estimated to be around $10.1 million in 2023 so it seems unlikely the Giants will increase their offer by much, especially considering that they have a bunch of other spending decisions to make this offseason including QB Daniel Jones.
Assuming he isn't tagged, Barkley would enter unrestricted free agency off of a 295/1312/10 season. He probably does deserve a contract near the top of the running back market, but finding the team that will pay that contract is never a simple proposition.