Roadside bombings across Iraq kill five US soldiers as violence continues.
Al-Zobaie was wounded in the attack at his home in Baghdad.
An aide said the suicide bomber was one of his own guards.
Mortar rounds slammed into a Shia area in Baghdad on Tuesday, killing at least four people and wounding 14, while sectarian clashes erupted for a fourth consecutive day south of the capital.
The mortar attack occurred in Abu Dasheer, a Shia enclave in the Sunni-dominated Dora neighbourhood in southern Baghdad. Police said those killed included two children, a woman and a man.
In Iskandariyah, 50km south of Baghdad, suspected Shia fighters broke into a Sunni mosque and planted explosives that damaged the gate and a fence, the police said.
Clashes broke out about an hour later, leaving four Sunni fighters and one Shia fighter wounded.
The clashes were followed by a mortar attack on a nearby Shia mosque. The rounds landed in a courtyard and did not damage the mosque, although a pedestrian was wounded.
A Shia man and his three sons also were wounded when a mortar round struck their house in Haswa, about 5km east of Iskandariyah.