The longer timeline is considered more likely, the source said.
The Giants have a bye in Week 11, which could give him eight weeks to recover and allow him to return for the last six games.
Barkley suffered the injury to his right ankle in the second quarter of the Giants’ 32-31 victory over the Tampa Bay Buccaneers on Sunday.
He was injured when his right ankle bent awkwardly while he was tackled by Bucs safety Mike Edwards. Barkley limped to the sideline, where he was immediately attended to by trainers.
Barkley entered Sunday second in the NFL with 227 yards rushing. He was averaging 7.8 yards per carry.
The second-year back had eight carries for 10 yards and four catches on five targets for 27 yards before exiting Sunday’s game with the injury.
Barkley, 22, was named Offensive Rookie of the Year last year after finishing with 2,028 total yards last season.
ESPN’s Jordan Raanan contributed to this report.