The longer time frame 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.
Giants coach Pat Shurmur wouldn't confirm a time frame for Barkley's possible return but said the Giants are confident in backup running back Wayne Gallman.
"We're just going to have to see first off. We certainly believe in Wayne Gallman and we certainly believe he will be able to step in and do his very best. And we'll just have to see at some point. We may have to add a running back behind him as we go forward here," he said Monday.
Gallman is the lone healthy natural tailback on the active roster. Fullback Eli Penny is a converted running back as well. Rookies Jon Hilliman and Austin Walter could be elevated from the practice squad.
"It's like any injury. It's very unfortunate to have Saquon out of the lineup. We all know what he brings to the team. But it's going to fall on all of us to move forward and do what we can to win games," Shurmur said.
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.
ESPN's Jordan Raanan contributed to this report.