Hassan's cousin, Ahmed Majid, said in an interview that Hassan's clothing caught fire and he struggled to jump off the truck, falling to the ground unconscious.
Military officials would not confirm whether Hassan was the same person shot by the soldiers. Majid and Raheem said they had been invited to testify at a military hearing Saturday.
Accounts of the incident by U.S. and Iraqi witnesses bear some similarities, but the two sides disagree on other aspects of the attack, including the extent of injuries suffered by the Iraqi.
Alban and Horne said in confessions that the man they shot was severely wounded and unlikely to survive. They said they watched him moan and writhe in pain until they could stand it no longer.
Sgt. Jacob E. Smith, an Army medic who helped treat the wounded Iraqis, testified that the victim's limbs were severely burned and his intestines were spilling out.
"Everything from his ribs to his hips was gone," Smith said. "He was in bad shape. He was going to die." Another witness said the man's spinal cord was exposed.
Majid said his cousin was unconscious and struggling to breathe, but his only injuries were burns. He said he pleaded with soldiers to help his cousin and his brother, who was still trapped in the burning truck. But when he tried to help Hassan, he said, a soldier pushed him away, saying, "Shut up and go!" Then the soldier shot his cousin, he said.
After the shooting, Majid said he saw two soldiers appearing to argue about the incident.
At a recent military hearing in Baghdad, an Army gunner, Spc. William Davis, testified that the two sergeants initially asked him to shoot the wounded Iraqi. Davis said he refused.
He described how an uninjured Iraqi male pleaded with Horne not to kill the wounded Iraqi on the ground.
"The guy was saying, 'No,' " Davis testified. " 'He's my brother! He's my brother!' "
According to Davis, Horne replied, "I understand, but he's gone."
Then, after consulting with the platoon leader and briefly debating what sort of weapon to use, Alban and Horne shot the Iraqi, according to testimony at the recent hearing. Horne told investigators that he believed the Iraqi was alive after Alban's first shot, so he shot him as well.
The platoon leader, Lt. Erick Anderson, remains under investigation but has not been charged. At a recent hearing, Anderson refused to testify, invoking his right against self-incrimination.
Majid said his family received a total of $7,500 in compensation for the deaths of Hassan, another cousin and his brother.
Efforts to find Iraqi witnesses were made only after an attorney for Alban criticized the government at a hearing for failing to do enough to track down the victim's family.
"The witnesses should have the opportunity to have their day in court," said Capt. Catherine Robinson, a military attorney appointed to represent Alban.
"Those Iraqis were there," she said. "They were in the dump truck. They know what happened. They know what happened to their cousins, brothers and whoever else was there."
Government attorneys said the lack of records at the Iraqi police department and the dangerous conditions in Sadr City had hampered their investigation." There are security issues," said Capt. Emily Schiffer, chief of the legal unit that is pressing the case. "Also there are a lot of holes in the accountability" of the Iraqi police.
Investigators said the lack of a body or autopsy report could also hinder the investigation.
"If we don't have a body, we really don't have a case," Special Agent Herman Vanderhorst, the lead criminal investigator, testified.
Horne, who joined the Army in 1999, has been negotiating a plea bargain with military officials, attorneys said.
The initial attack by U.S. soldiers on the garbage truck is also under investigation to determine whether it was appropriate to open fire without warning. Military officials declined to provide details, saying an inquiry was underway.
Iraqis caught in the attack said they wanted an explanation.
"The Americans were acting like real cowboys," Raheem said. "I was just doing my job. There was nothing to suggest we were armed. Why did they open fire on us? I'm still waiting for an answer to that question."
Times special correspondent Raheem Salman contributed to this report.