"[Witten] will be day-to-day all week, but he's not a lock [to play] at this point," a source said.
Witten insisted he was fine after Sunday's victory over the Philadelphia Eagles, referring to the injuries as "typical football stuff."
Witten, who is known for his high pain tolerance, missed one game early in his career with a broken jaw. He also insisted on playing in the season opener against the New York Giants after rupturing his spleen in the preseason in 2012.
Witten's 133 consecutive starts rank first among active NFL tight ends, and only Houston Texans punter Shane Lechler (208) has played in more consecutive games than the 189 straight he has appeared in for the Cowboys, according to ESPN Stats & Information.
The Cowboys will be without quarterback Tony Romo and wide receiver Dez Bryant, both of whom have long-term injuries that challenge Dallas' coaching staff to create offense and score points.
ESPN Cowboys reporter Todd Archer contributed to this report.
Jason Witten's consecutive-start streak in jeopardy
ESPN Cowboys reporter Todd Archer discusses whether Jason Witten will be on the field against the Falcons.