But New Jersey state police disagreed, saying who owned the gun had yet to be determined.
According to USA Today Sports, lawyer Anthony Fusco said he received records related to Goodson's case that showed the driver, Garant Evans, 31, of Roselle, N.J., claimed ownership of the 45-caliber Taurus Millennium gun after the arrest.
Provided by Fusco to USA Today Sports, a portion of the New Jersey state police investigation report filed by the responding trooper stated: "Upon briefly speaking to Evans, he stated that the gun was his but refused to answer questions regarding where it came from or where he acquired the gun. I attempted to have Evans sign a copy of the consent form which was read to him at the scene and which he gave verbal consent to search the vehicle. Evans refused to sign and same was noted on the consent form."
In response, New Jersey state police spokesman Sgt. Adam Grossman told NFL.com that investigators still are trying to determine who had the gun. One factor is the intoxication of the men at the time. The police also are awaiting records that will show to whom the gun was legally registered, Grossman said.
"At the time of arrest, it could not be determined the ownership of the firearm, so they were both charged," Grossman said.
Grossman added that the investigation is ongoing.
The attorney for the driver also weighed in, telling the New York Post: "Any statements allegedly made by Mr. Evans at the time of his arrest will be the subject of future hearings."
Goodson and Evans were arrested about 3 a.m. on May 17 after state police said they found Goodson and Evans possibly intoxicated inside an SUV stopped on the interstate with marijuana and a loaded gun inside.
The charge for unlawful possession of a handgun carries a minimum three-year prison sentence if convicted.
Fusco has stated all along that both the gun and marijuana weren't Goodson's. However, marijuana was found in Goodson's pocket, according to a police affidavit obtained by USA Today Sports.
Goodson, a fifth-year player, signed a three-year, $6.9 million contract with the Jets in the offseason.
Goodson also reportedly is facing a paternity lawsuit. He hasn't spoken to the media since the arrest
The Jets have stated they're gathering information on the incident.