NEW ORLEANS – NFL Commissioner Roger Goodell rejected the appeals of four players suspended in connection with the league’s bounty investigation of the New Orleans Saints.

In a ruling handed down on Tuesday, Goodell told Jonathan Vilma, Anthony Hargrove, Will Smith and Scott Fujita that he retains “the inherent authority to reduce a suspension should facts be brought to my attention warranting the exercise of that discretion.”

“The record confirms that each of you was given multiple chances to meet with me to present your side of the story,” Goodell said. “You are each still welcome to do so.”

Vilma is suspended for the entire 2012 season, while Smith will miss four games. Hargrove, now with Green Bay, was suspended eight games, while Fujita, now with Cleveland, was suspended three games.

The NFL Players Association issued a statement saying it will continue to pursue all options.

“The players are disappointed with the League’s conduct during this process,” the statement said. “We reiterate our concerns about the lack of fair due process, lack of integrity of the investigation and lack of the jurisdictional authority to impose discipline under the collective bargaining agreement.

“Moreover, the Commissioner took actions during this process that rendered it impossible for him to be an impartial arbitrator.”

Goodell said he did not take his initial decision lightly and also points out that players did not help their cause by refusing to participate fully in the appeal process.

Vilma and his attorney, Peter Ginsberg, walked out of the hearing early after Ginsberg raised his objections to the NFL’s handling of the entire investigation. The three other players, who were represented by NFL Players Association attorneys, sat through the hearing to observe the NFL’s presentation of evidence, but in protest refused to present any evidence or witnesses of their own, and did not question the NFL investigators who were present at the hearing.

The NFL issued a report in March saying that league investigators determined the Saints ran a bounty program from 2009 to 2011 that offered improper cash payments for hits that injured targeted opponents.

The league has said former Saints defensive coordinator Gregg Williams ran the program, and that general manager Mickey Loomis and head coach Sean Payton failed to put a stop to it despite warnings from the NFL at the end of the 2009 season.

The NFL has suspended Williams indefinitely and Payton for the whole season. Loomis is suspended the first half of next season, while Saints assistant head coach Joe Vitt was suspended for the first six games. Goodell also docked the Saints second-round draft picks this year and next and fined the club $500,000.

Brees gets win over franchise tag

Drew Brees gained additional leverage in his protracted contract talks with New Orleans when an arbitrator in Philadelphia ruled in Brees’ favor in a dispute over how much the Saints would have to pay the star quarterback if they applied the franchise tag to him again in 2013.

The Saints have already used the tag on the former Purdue star for 2012, meaning he can’t negotiate with another team and could be forced to settle for a one-year, $16.3 million deal if he cannot reach a new long-term deal by a July 16.

System arbitrator Stephen Burbank’s ruling said if New Orleans tries to tag Brees a second year in a row, he would be entitled to a 44 percent raise to about $23.5 million because it would be his third-career franchise tag. The NFL had argued Brees would be due a 20 percent raise because it would have been only his second franchise tag with one team.

Burbank, however, ruled that while the NFL’s CBA has some ambiguity on the matter, it is clear that the overarching purpose of the language regarding multiple franchise tags is meant to protect players from being denied their rights to free agency for an undue length of their careers.

Brees has so far skipped the Saints’ offseason practices while holding out for a new long-term contract with New Orleans.