Members of the Huwaitat tribe accuse the Jordanian government of "covering up the truth" and say officials are hiding details that should exonerate Maarik al-Tawaiha.
The 39-year-old was convicted in June of killing of three American Special Forces officers - Matthew C Lewellen, 27, Kevin J McEnroe, 30, and James F Moriarty, 27.
The three were part of a CIA programme training members of Syrian opposition groups in Jordan when they were killed at King Faisal Airbase at al-Jafr, in southern Jordan, on November 4.
Sayel Abu Tayeh, a leader of the Huwaitat tribe and a royal court adviser on tribal affairs, said in a statement early this week that Tawaiha followed the rules of engagement as he believed the base was under attack by unknown assailants.
The CIA used the King Faisal Airbase to train Syrian fighters battling the Syrian government of
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The Tawaiha family's version of events that led to the shooting of the Americans is sharply in contrast with the official account of the incident.
Abu Tayeh, a close relative of Tawaiha, accused key figures in the Jordanian government of a "cover-up" and of using his relative as a "scapegoat" in order to deflect American pressure and cover their own security failure.
"What happened on that day was a mix-up that led to the tragic shooting," he said.
"We are saddened that Americans were killed in that incident, but Maarik was no murderer."
Al Jazeera could not confirm many of Abu Tayeh's claims regarding the current or former Jordanian government officials.
But Al Jazeera has obtained several Jordanian court documents, including Tawaiha's own statement and those of witnesses.
Those witnesses said they heard a gun shot being fired, without specifying by who or from where, which triggered the chain of events that later followed.
They also described a shootout between the soldiers defending the base's gate and "unknown assailants" riding in cars approaching the base and shooting at it.