Attorneys for Tom Brady want the full U.S. Court of Appeals for the Second Circuit to rehear his case. (Michael Nagle/Bloomberg)

The seemingly never-ending Deflategate saga advanced to yet another phase Monday when Tom Brady’s legal team asked for a rehearing before a federal appeals court of the ruling that reinstated the quarterback’s four-game suspension.

Attorneys for Brady and the NFL Players Association argued that the U.S. Court of Appeals for the Second Circuit should hear the case and restore Brady’s eligibility to remain in the New England Patriots’ lineup without interruption.

“The panel decision stands in stark conflict with fundamental rules of labor law and undermines the rights of union members and employers alike,” the request filed on Brady’s behalf says. “This Court should grant rehearing.”

Legal analysts called Brady’s remaining legal options long shots after a three-judge panel of the same appeals court ruled in April that NFL Commissioner Roger Goodell had exercised his authority appropriately under the sport’s collective bargaining agreement by upholding Brady’s suspension on appeal last year.

If Brady fails to get his suspension overturned by the full appeals court, his next option would be to appeal to the U.S. Supreme Court.

“The facts here are so drastic and so apparent that the court should rehear it,” Ted Olson, an attorney for Brady, said in a televised interview with ABC.

Seven of 13 active judges of the federal appeals court would have to agree to rehear the case. The appeals court’s chief judge, Robert A. Katzmann, dissented in the April decision by the three-judge panel. Some observers have speculated that might bolster the chances that the full appeals court will agree to rehear the case.

But legal analysts said it is rare for such a request for an en banc hearing to be granted.

“Courts of appeal typically reserve en banc rehearing for cases of broad significance in which reconsideration of the original panel ruling seems necessary to correct error or to clarify some point of law,” Boston College law professor Alfred C. Yen said via e-mail. “In this case, the original Second Circuit opinion follows fairly well settled law, with the possible exception [of] Commissioner Goodell’s alleged bias as an arbitrator. The entire Second Circuit could conceivably interpret the original panel decision as one of broad significance, but I would think this fairly unlikely.”

Yen said a potential petition to the Supreme Court by Brady would also unlikely be heard. He added in a subsequent phone interview he doesn’t regard Katzmann’s dissent in April as a “game-changer” indication that the full appeals court will agree to hear the case, noting that “judges are notoriously independent.”

The NFL contends that Goodell is empowered by the CBA to act as he did and the courts should not be used to review the facts of the case. The NFLPA argues that Goodell exceeded his authority and did not follow a fair and proper process.

The petition filed Monday says the case “arises from an arbitration ruling by NFL Commissioner Roger Goodell that defies the rule of law.”

Brady’s request argues that Goodell improperly affirmed Brady’s suspension based on a different set of findings than those originally resulting in Brady’s suspension. The petition says that Goodell “upheld the suspension based on Brady’s supposedly having ‘participated’ in a conspiratorial ‘scheme,’ his having given gifts to the employees who allegedly deflated the footballs, and his ostensible obstruction of the investigation. … Neither the Wells Report nor the disciplinary order on appeal made any of these findings or purported to impose discipline on these grounds.”

It also contends that Goodell failed to give proper consideration to the argument that Brady had committed an equipment violation which should have resulted in merely a fine.

The April ruling by the three-judge panel reversed last year’s decision by U.S. District Judge Richard M. Berman that overturned the suspension of the Patriots’ four-time Super Bowl-winning quarterback.

“The league has a history of being bullies,” DeMaurice Smith, the executive director of the players’ union, said in a televised interview with ESPN. “And today we’re filing a brief because the commissioner decided to be a bully when it came to the fair hearing of a player.”

The legal tussle over Brady’s use of under-inflated footballs in the first half of the AFC title game at the end of the 2014 season drags on. Berman’s decision enabled Brady to play all of last season for the Patriots, with the NFL’s appeal pending, as they reached the AFC championship game but lost at Denver.

The union and Brady potentially could seek a stay of the suspension now to keep Brady eligible to play, at least temporarily, in the upcoming 2016 season while the legal process continues to unfold.

But Yen said that Brady and his lawyers cannot necessarily count on being granted a stay.

“Realistically his best chance of getting into those first four games might be to reach some sort of agreement with Commissioner Goodell,” Yen said.

The Patriots are scheduled to open their season Sept. 11 at Arizona. If Brady serves his four-game suspension at the outset of the season, he also would miss games at home against the Miami Dolphins, Houston Texans and Buffalo Bills before he would be eligible to play Oct. 9 at Cleveland. Third-year backup Jimmy Garoppolo would be in line to open the season as the Patriots’ starter if Brady is serving his suspension.

Goodell has said that the legal fight between the league and union is not so much about Brady as it is about the commissioner’s disciplinary authority under the CBA. Smith expressed similar sentiments Monday, although from a far different perspective.

“We’re the union and we represent all of our players,” Smith told ESPN. “We stay in touch with our players and certainly Tom understands the importance of a union that fights for every player’s rights. And I think that he is certainly a player who understands that this case is bigger than he and every fight that we have with the National Football League is a fight with all of the players and for players in the future.”

Negotiations between the league and union aimed at reaching an accord on the sport’s system of player discipline — and Goodell’s role in it — recently unraveled. The union had been seeking independent arbitration of disciplinary measures taken by the NFL in cases under the sport’s integrity-of-the-game rules, such as Brady’s, and under the personal conduct policy.