Alaa Hassuna, 23, was shot in the Bait Lahya area on Thursday shortly after leaving the family home in a nearby beduin village around dawn.
His wife and another passenger in the vehicle were wounded, the sources said.
Hassuna died instantly, a Palestinian security official said.
However, the Israeli army said troops fired near the man's vehicle because he was speeding towards them and they feared an attack was imminent.
As a result of the gunfire - which the army said did not hit the vehicle - it veered off course, and it is possible the man died as a result of the crash, the Israeli army said.
Shot in the eye
The military's version was contested by Mahmud al-Asali, director of the Bait Lahya hospital, who said Hassuna died of fatal bullet wounds.
"He was shot in the right eye and the bullet exited from the back of the brain. There were no scratches to show that he had been in an accident," he said.
Hassuna's wife sustained a bullet wound to the hand while the second passenger was slightly injured by a bullet wound to the leg, al-Asali added.
The Israeli army added that its arrest operation in Bait Lahya was "successful" and that a "fugitive" had been taken into custody.
Also on Thursday, a Palestinian was shot dead by Israeli troops during an army incursion into a refugee camp in the central Gaza Strip, Palestinian medical sources said.
Said Abd al-Salaam, 22, was hit in the chest when troops swarmed into the Buraij refugee camp, which lies between Gaza City and Dair al-Balah, they said, adding that several others were wounded during the raid.
Israel frequently kills fighters by
firing missiles at their vehicles
An army spokeswoman said there had been an overnight arrest operation just east of the camp, during which an armed Palestinian approached the troops and opened fire.
"They returned fire and identified a hit," she said.
In another incident, an Israeli helicopter fired missiles at a Palestinian car in the Gaza Strip and witnesses said they saw smoke rising from the vehicle.
"We attacked a suspicious car in an open area south of Kfar Darom and identified a hit," an Israeli military source said. Kfar Darom is an illegal settlement.
The car was standing in an empty field when the helicopter
launched the attack, the source said. Palestinian medics said it appeared no one was hurt.
Israel Radio said the missile strike was not an assassination attempt against Palestinian fighters but an attack on a suspected car bomb.