Cowboys coach Jason Garrett said on Tuesday that Smith would not practice because of a neck injury, and Garrett expects his availability for Thursday to be a game-day decision.
Smith did not play in Dallas' victory over the Washington Redskinson Thanksgiving Day, although he was on the active roster for the game. Cameron Fleming occupied the left tackle position for the Cowboys in that game.
If Smith sits out again against the Saints, he would have quite a bit of time to recover from his injury before the Cowboys play their next game against the Philadelphia Eagles on Sunday, Dec.9.
Smith has started 115 NFL games in his career, including starts in all 10 games in which he has played this season.
For the Saints, left tackle Terron Armstead has missed the past two games with a shoulder injury, and he is not expected to play against the Cowboys, NFL Network's James Palmer reported.
Armstead has started all nine games in which he played this season.
Jermon Bushrod, who was a Pro Bowl selection in 2011 and 2012, will continue to fill the spot occupied by Armstead.