Barkley and the Giants have until 4 p.m. ET Monday to finalize a deal. Otherwise, Barkley is not eligible to be re-signed until after the 2023 season.
As the deadline nears, the two sides remain at a stalemate, sources told ESPN.
Barkley finished fourth in the NFL with 1,312 rushing yards and 10 touchdowns last season. He also tied for the team lead with 57 receptions.
The No. 2 pick in 2018 has made it clear that the franchise tag, which comes with a $10.1 million salary this season, is undesirable. Barkley said at his Athletes Making Progress Together football camp last month that if he didn't get a long-term deal by July 17 that he would at least contemplate sitting out. He also added that he hadn't put much thought into it at the time.
This entire contract situation has not sat well with Barkley.
"Me getting tagged, was I upset about it? Nobody wants to get tagged," Barkley said last month. "To sit here and say I was frustrated, I was mad, I was upset, what really got me upset was the stories that got leaked out, how misleading they were and how untruthful they were.
"I feel it was trying to paint a narrative of me, a picture of me, that is not even true. Not even close to being the truth."
Barkley has found trying to negotiate a new contract an eye-opening experience.
"Just come to a reality that this is a business," he said.
Five days still remain for the two sides to reach agreement on a new deal. If it doesn't happen, Barkley will have to make a major decision.
The last player to sit out an entire season was running back Le'Veon Bell in 2018.
History has not reflected kindly on that decision. Bell signed with the New York Jets the next season but never regained his previous form.