Vilma, named by the NFL as a ringleader of the operation, appealed his season-long suspension while the NFL Players Association notified the league it was reserving the appeal rights of Saints defensive end Will Smith and those of ex-Saints Anthony Hargrove and Scott Fujita.
All four players and the union argue that no appeal should be heard before NFLPA grievances are resolved questioning NFL Commissioner Roger Goodell's authority to discipline the players and to hear their appeals.
Vilma's appeal also states the NFL has not presented any evidence linking him to a system in which players were paid to injure opponents. It asks the league to provide a wide range of documentation, including witness statements and the names of those witnesses.
Vilma received the harshest of the suspensions, followed by Hargrove (eight games) Smith (four games) and Fujita (three games). Monday was their deadline to appeal.
Hargrove now is with Green Bay. Fujita, a member of the NFLPA's executive committee who has played for Cleveland since 2010, made his first public statement since all four suspensions were handed down last Wednesday. Like Vilma and Smith before him, Fujita denied involvement in a "bounty pool" and challenged the league's findings.
"I disagree wholeheartedly with the discipline imposed," Fujita said in a statement emailed to The Associated Press. "I've yet to hear the specifics of any allegation against me, nor have I seen any evidence that supports what the NFL alleges.
"I look forward to the opportunity to confront what evidence they claim to have in the appropriate forum," Fujita continued. "I have never contributed money to any so-called 'bounty' pool, and any statements to the contrary are false. To say I'm disappointed with the League would be a huge understatement."
The players' manoeuvrs came on the same day copies of a sworn statement by Hargrove were obtained by media including the AP.
Hargrove's statement describes how he was instructed by ex-defensive co-ordinator Gregg Williams and current New Orleans assistant head coach Joe Vitt to deny the existence of a bounty program to NFL investigators.
The interpretation of his statement is a matter of debate, however.
The NFL has said that Hargrove's words acknowledge the existence of a bounty program and show that Hargrove initially lied to NFL investigators about it. The union notes that Hargrove's statement does not say that he lied to anyone, nor does it state that he or any other Saints participated in a bounty program.
The argument is one of many that are bound to play out on appeal, once the union's jurisdictional challenges have been resolved.
The union's grievances argue that Goodell is prohibited from punishing players for any aspect of the case occurring before the current collective bargaining agreement was signed last August. It argues that a CBA system arbitrator, and not Goodell, has the authority to decide player punishment under such circumstances, as well as rule on any appeals.
Vilma's latest filing not only reiterates those positions but also states that the NFL still has not provided "a single piece of evidence" to the Saints defensive captain to justify his suspension.
"To be able to share, discuss and analyze the supposed evidence that has been gathered is a fundamental cornerstone of a fair and just process, and a vital prerequisite to uncovering the truth," wrote Vilma's attorney, Peter Ginsburg. "Indeed, the failure of the NFL to conduct itself in a just manner has compromised the process and resulted in erroneous and damaging conclusions."
Vilma's legal team now wants to see if there really is evidence such as account ledgers of improper cash bonuses, payment slips or other documents or emails showing Vilma pledged, made or received bounty payments.
Vilma also asks to review any video or audio evidence that the NFL has, including video from games or any statistical analysis of Vilma's on-field performances.
Last week, former U.S. attorney Mary Jo White, who was hired by the NFL to evaluate its bounty investigation, said there was evidence from "multiple independent sources" that shows players received payments for going after particular opponents.
The NFL has said its investigation included 18,000 documents comprising nearly 50,000 pages.
White said the NFL has shared ample evidence with suspended players and the NFLPA, and she also said that concealing the identity of witnesses is important in terms of not only protecting those who help investigations but encouraging more to step forward in the future.
The Saints already have been punished heavily in connection with the probe.
Head coach Sean Payton has been suspended the entire 2012 season, while suspensions of eight games were handed down to general manager Mickey Loomis and six games to Vitt. The club also was fined $500,000 and docked two second-round draft choices this year and next. Meanwhile, Williams, who admitted to running the bounty program from 2009-11, has been suspended indefinitely. Williams was hired by the St. Louis Rams in the off-season.