The Dallas Cowboys have concerns after Odell Beckham Jr.'s physical that his recovery from a torn left ACL in the Super Bowl has not progressed enough to ensure he would play before mid-January, a source tells ESPN's Ed Werder.

There is a possibility that signing Beckham, who closed out his two-day visit with the Cowboys on Tuesday, would have no benefit until the 2023 season, the source added.

Beckham underwent a physical on Monday, met with Cowboys owner Jerry Jones and his family, and spent the night with Micah Parsons and Trevon Diggs at the Dallas Mavericks' game against the Phoenix Suns. Asked at halftime what the Cowboys' chances of signing him were, Beckham told ESPN's Tim MacMahon with a big smile, "It's a good possibility."

Jones remained noncommittal during an appearance on 105.3 The Fan in Dallas on Tuesday, but did say that Beckham playing this season is "paramount" to a deal possibly getting done without ruling it out altogether.

"It's a lot different if you play one play or if you played or are available for a week and that week be the Super Bowl week, than if he's available the next week," Jones said. "And so everything in between. I would say that's a point of discussion -- a player's own belief of where he is in becoming ready to play is real big."

Beckham, who has also visited with the New York Giants and Buffalo Bills, was expected to meet Tuesday with Cowboys' player leadership council that includes Dak Prescott and 10 other veterans.

According to NFL Players Association figures, the Cowboys have $6.3 million in salary-cap room, which would be enough to add Beckham; however, a multiyear deal would further complicate a dicey salary-cap situation in 2023. The Cowboys have 19 players set to become unrestricted free agents, including key figures like Dalton Schultz, Tony Pollard, Leighton Vander Esch and Donovan Wilson. They can also open up contract-extension talks with CeeDee Lamb and Diggs for the first time.

"I just think this is an opportunity that is rare because of his availability," Jones added. "But one way or the other, there's not a failure here. I'm talking about the actual signing or not signing. There's not disappointment here.

"If this thing works, it will improve this team now. It will improve it this year. If we don't have that, then we're in a positive situation as well. It has to improve us now. You know that's a pretty tall order. And it has to improve us in a way that makes sense for us as we look not only this year but as it impacts the years to come relative to the financial aspect of it."

ESPN's Todd Archer contributed to this report