The witnesses said that Jilani was standing on the side of the road when Israeli police officers started firing indiscriminately, killing him and seriously injuring a young woman.
But Micky Rosenfled, the Israeli police spokesman, said officers opened fire after the man whose car hit them attempted to flee on foot.
Israeli sources said the two officers were lightly injured after being struck by the car.
Amin Abu Ghazaleh, a doctor with the Palestinian Red Crescent Society, said two other men were also wounded in the incident.
Shumel Ben-Ruby, an Israeli police spokesman, said the car Jilani was driving also carried a passenger who apparently had been injured in a stone-throwing incident.
Giving a different account of the incident, two witnesses told the Israeli daily Haaretz that stones had been thrown at the police officers, one of which struck Jilani's car.
This caused Jilani's car to veer off its path and strike the police officers.
'Crime of occupation'
Reacting to the news of Jilani's death, the Palestinian Authority (PA) said in a statement that it strongly condemned "the crime of the occupation force in Jerusalem".
The statement said the PA had "suspicions about the Israeli story" and demanded "a neutral investigation into the incident".
Speaking to Al Jazeera, a family friend described Jilani as a "very family-oriented man, a loving husband and a devoted father" to his three daughters.
"His wife told me that before her husband left the house that morning, they had discussed taking the kids out some place fun that afternoon," she said.
"This was no premeditated attack."
The shooting that killed Jilani occurred at a police checkpoint in Wadi al-Joz, a neighbourhood in East Jerusalem which Israel captured during the 1967 Middle East war and subsequently annexed.
Israeli control over Wadi al-Joz is not recognised by the international community and Palestinians want East Jerusalem as the capital for their future state.
Tensions between Israelis and Arabs have been high in East Jerusalem.
Israeli police have announced a limited-access policy for Palestinian men under 40 travelling from East Jerusalem to al-Aqsa mosque for Friday prayers.
Israeli security forces were deployed across East Jerusalem in large numbers on Friday in anticipation of possible unrest.