The person spoke to The Associated Press Friday on condition of anonymity because the team has not announced any details of Bryant's medical exam.
Bryant's reaction was muted, referring to the injury as "the ultimate test".
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Bryant, who played his entire career in Dallas before this week, was cut by the Cowboys this offseason in what was at the time considered a surprising move.
Seeking salary cap relief, the Cowboys released Bryant after last season with two years remaining on the receiver's five-year, $70 million contract.
For Bryant, the injury continues a tumultuous year that began with a messy divorce in Dallas.
Dez Bryant signed with the New Orleans Saints earlier this week and there was a lot of excitement about him returning to the National Football League after a long hiatus away from the game.