Umenyiora injured his ankle -- the team has not said which one -- in a loss in New Orleans on Nov. 28 and has missed the past four games.
While Umenyiora has been sidelined, second-year player Jason Pierre-Paul emerged and was selected to the Pro Bowl. Pierre-Paul doesn't believe it's automatic he should return to the bench, and told NFL.com's Steve Wyche he should start Sunday regardless of Umenyiora's status.
"It's my birthday," Pierre-Paul cited as his rationale to start.
Pierre-Paul, who will turn 23 on New Year's Day, did not say he should be the full-time starter if Umenyiora was healthy. But he did say -- with a smile -- he should be "for this game."
Pierre-Paul did acknowledge the added depth would keep him fresher throughout the game.
"That rest will help me a lot and it will get me going," said Pierre-Paul, who was selected the NFC defensive player of the week on Wednesday for the second time in three weeks. "I was playing like 80 snaps a game and that is a lot. You don't really think about it, you just have to go."
Giants coach Tom Coughlin said wide receivers Hakeem Nicks (hamstring) and Mario Manningham (knee) were held out of practice, but was optimistic both would play against the Cowboys. Nicks was injured last week against the New York Jets but played through the injury, according to Coughlin.
As expected, running back Ahmad Bradshaw (foot) also sat out practice Wednesday.
The Associated Press contributed to this report