According to the filing, Brady's legal team, led by former U.S. Solicitor General Ted Olson, petitioned for a panel rehearing or a rehearing en banc. In an en banc hearing, all 13 judges on the court would rehear the case.
"The facts here are so drastic, and so apparent, that the court should rehear it," Olson told ABC News in an interview that aired Monday morning.
Appearing on NFL Total Access on Monday, Gabe Feldman, director of the Tulane Sports Law Program, characterized the en banc rehearing request as "a long shot" to be accepted by the court.
In the filing, lawyers for Brady and the NFL Players Association argue a "rehearing is warranted because the (appeal) panel opinion conflicts in two key respects with decisions of the Supreme Court and decisions of other circuits."
The petition cites Judge Robert A. Katzmann in his dissenting opinion last month when he wrote NFL Commissioner Roger Goodell "exceeds his limited authority under the CBA to decide 'appeals' of disciplinary decisions." Brady's lawyers also argue "the Commissioner completely ignored the collectively bargained schedule of penalties for equipment-related violations." Citing Katzmann again, the petition argues that those collectively bargained equipment violations should have "limited the discipline" or "reduced" the sanctions against Brady.
"The divided panel of the Second Circuit reached erroneous legal conclusions under an unfair and unjust standard," Olson said in a statement. "The decision and the standards it imposes are damaging and unfair -- not only to Tom Brady -- but to all parties to collective bargaining agreements everywhere. Commissioner Goodell cannot sit as an appellate arbitrator and then affirm the league's initial disciplinary decision based upon a new theory and imagined evidence and pretend to be an unbiased decision-maker."
DeMaurice Smith, executive director of the NFLPA, argued in a statement that Brady "was not afforded fundamental fairness and due process as guaranteed by the collective bargaining agreement and case law.
"We also know that the NFL propped up a now completely de-bunked 'independent' report with a made-up standard as the basis for his suspension," Smith said.
There is no timeline for the court to decide whether or not it will rehear the case. If the court accepts the petition, there would be a stay of Brady's four-game suspension until the new ruling, and he'd be eligible to play.
If the court chooses not to rehear the case, Brady could then petition the U.S. Supreme Court and ask for a stay of the second circuit decision. If the second circuit doesn't grant a stay, Brady's lawyers can ask for a stay of the decision from the Supreme Court, which would be decided by Justice Ruth Bader Ginsberg. If Ginsberg granted a stay or the court decided to hear the case, Brady almost certainly would be eligible to play the entire 2016 season due the Supreme Court's schedule.
Feldman said the chances of the Supreme Court deciding to hear the case would be very low.
Brady was suspended four games by the NFL in May 2015 after an NFL-commissioned investigation conducted by attorney Ted Wells concluded it was "more probable than not" that Brady was "generally aware" of Patriots attendants deflating footballs prior to the AFC Championship Game against the Indianapolis Colts on Jan. 18, 2015.
The four-time Super Bowl winner's suspension was initially nullified by U.S. District Judge Richard Berman last September, a week before the first game of the regular season. An appeals panel reversed Berman's decision in April, stating in the majority opinion that Goodell "properly exercised his broad discretion under the collective bargaining agreement" in issuing Brady's suspension.
The Patriots open the season on the road against the Arizona Cardinals on Sunday Night Football. New England then plays three straight home games against the Miami Dolphins, Houston Texans and Buffalo Bills. If Brady were to end up serving his suspension this season he would be eligible to make his 2016 debut Oct. 9 on the road against the Cleveland Browns.