The Saints’ Jimmy Graham may have lost his recent grievance hearing in front of an NFL arbitrator, but he isn’t quite ready to give up the fight.

He has served notice that he is reserving his right to appeal the decision, according to an anonymously-sourced report from The NFL Network on Monday.

Nonetheless, simultaneous reports from ESPN and Pro Football Talk said there’s optimism the appeal notice won’t prevent Graham and the Saints from striking a new long-term contract by a deadline of 3 p.m. local time Tuesday, which would render the review of the arbitrator’s decision moot.

After July 15, Graham in 2014 can only either sign a one-season franchise tag offered to him by the Saints or sit out the year.

Graham’s decision to pursue an appeal could be seen as an indication that there’s still work to do in negotiations for a long-term contract. But in no way does it bar the Saints and Graham from agreeing to a deal by 3 p.m. Tuesday, nor does it affect that deadline.

It’s been a virtual assumption that Graham would not take a contract that didn’t offer him a good bit more than the six-year, $54 million deal (more than $13 million of which were guaranteed) that made New England’s Rob Gronkowski the best-paid tight end in NFL history.

None among the Saints, Graham’s camp and the NFL players have responded to requests for comment throughout the day Monday.

Monday’s reports were the latest twists in a quasi-legal clash between Graham and the Saints dating back to when New Orleans handed the star player a franchise tag that prevented him from becoming an unrestricted free agent when the four-year contract he was given as a rookie in 2010 expired in March.

The tag the Saints gave Graham was worth more than $7 million for the 2014 season and designated him as a tight end. But because he lined up mostly as a slot receiver in 2013, when he led the Saints with 1,215 receiving yards and the NFL with 16 touchdown catches and was named Associated Press First Team All-Pro, Graham filed a grievance through the league players union contending that he should get a wide receiver franchise tag worth more than $12 million.

NFL system arbitrator Stephen Burbank, who presided over a hearing concerning the grievance, ultimately ruled that Graham was a tight end. Burbank arrived at the decision on July 2 after hearing testimony from Saints coach Sean Payton and General Manager Mickey Loomis that — among many other things — Graham was defended by opponents and treated by the team like tight ends normally are.

Graham was given 10 days from the date of the decision to appeal it. Because the 10th day was this Saturday and the league office was closed, the deadline extended until Monday, NFL Network reported.

Unless expedited, Graham’s representatives have between 15 days and no more than 25 from the date of the decision to lodge a brief in support of an appeal.

Those advocating for Graham’s opponents in the matter — the Saints and the NFL — would have as much time to respond in writing to any appellate brief. Oral arguments would be set between five and 10 days after that. And a written decision on the appeal would be issued within 30 days of oral arguments, though the schedule could be accelerated for “good cause,” according to the NFL collective bargaining agreement adopted in 2011.

A three-person panel oversees all appeals the nature of Graham’s. It is comprised of James Oldham, a Georgetown University law professor; Richard Holwell, a U.S. District Court judge in the Southern District of New York from 2003 to 2012; and Fern Smith, a retired U.S. District Court judge who served in the Northern District of California from 1988 to 2005.

Oldham is an expert in contracts, labor and employment law, legal history and torts. Holwell was appointed to the bench by President George W. Bush and eventually resigned his post to enter a private legal practice. Smith received her bench appointment from President Ronald Reagan.