Elliott attended the roughly two-hour hearing in NY on Monday, a day after rushing for 150 yards and two touchdowns in the Cowboys' 33-19 win at Washington.
If Judge Failla denies the emergency motion, the NFLPA plans to go to the Second Circuit asking for the injunction pending appeal in that court.
Judge Failla denied a request for a preliminary injunction on Monday night after hearing arguments from the National Football League and NFLPA. The NFLPA requests that Failla rule on its motion by 7 p.m. ET Wednesday, at which point it will seek relief from the appeals court.
A verdict on the appeal should come within the next couple of days.
Elliott would be eligible to return on December 17 for a game against the Oakland Raiders.
Draymond Green and Bradley Beal tangle during heated contest
He was caught on video while walking to the locker room screaming that the National Basketball Association was "out to get me". Both Green and Beal received fighting technicals and were immediately ejected, with Golden State going on to win 120-117.
The Cowboys now sit second in the NFC East at 4-3.
Elliott was suspended by NFL Commissioner Roger Goodell in August following a year-long investigation into domestic violence allegations made by Tiffany Thompson, his former girlfriend.
Elliott attended the roughly two-hour hearing in NY on Monday, a day after rushing for 150 yards and two touchdowns in a 33-19 win at Washington. That left Elliott free to play games against the San Francisco 49ers and Washington Redskins. He's been fighting the suspension in court ever since.
Judge Amos Mazzant granted a preliminary injunction that allowed Elliott to play until the resolution of the lawsuit.
Unlike three federal judges before her, Failla rejected most of those claims and backed the NFL's contention that it followed the collective bargaining agreement in suspending Elliott, and that those procedures were supported by federal labor law.
It also states that the "irreparable harm" the NFLPA argued Elliott would suffer is "either speculative or deserving of a monetary award rather than an injunction". Elliott appealed, and the NFL's arbitrator, Harold Henderson, upheld Goodell's decision. The league concluded he violated its personal conduct policy, which mandates a six-game suspension for first-time domestic violence violations.