The newspaper said the deal was for one year at $2.5 million. Elam played under Dallas defensive coordinator Rob Ryan in Cleveland for the past two seasons when Ryan was the defensive coordinator there.
After staying on the ground briefly, Newman slowly walked across the field to the sideline, where he crouched before being tended to by a trainer. He left the field shortly after that.
An MRI was planned to determine the severity of the injury.
The Cowboys did not comment about the seriousness of the injury, but a team source told the Star-Telegram late Wednesday he hopes Newman will be back in time for the season opener.
Newman, who is going into his ninth NFL season, missed all four preseason games in 2008 because of a groin injury. He returned for the season opener, but later missed five games because of a sports hernia and played a career-low 10 games that season.
Newman started all 32 games the past two seasons. He didn't miss a game last season despite dealing with ankle and rib injuries.
The Associated Press contributed to this report.