Reuters news agency reported that Radhi and one of his brothers, who is a junior member in the Development and Reforms Movement, were killed in the attack.
In Sunday's other violent incidents, three people were shot dead by unknown assailants from a car in Saadiya, a town in Diyala province, 120km northeast of Baghdad.
Four other people were wounded in the shooting, according to a Diyala security official.
In Baghdad, a roadside bomb went off near a government-backed patrol in the southwestern Jihad district, wounding two people, police said.
In Mosul, 390km north of the Iraqi capital, clashes between police and anti-government fighters killed one civilian and wounded another in the city's west, police said.
A Sunni Muslim leader who on Friday night celebrated Allawi's win by passing out candy to well-wishers was killed by a sniper in a Baghdad neighbourhood on Saturday morning, police and hospital officials said.
One of the deadliest attacks in months took place on Friday in Diyala when two bombs struck a crowded market, killing 59 people and wounding 73 more.
Violence has dropped dramatically across Iraq since its peak in 2006 and 2007, but attacks remain common, especially in Baghdad and Mosul.
Sunday's attacks came just two days after Iraq's election commission announced complete results from the March 7 vote in which Allawi's Iraqiya narrowly edged out the State of Law bloc led by Nouri al-Maliki, the incumbent prime minister.
Allawi's leadership bid
Allawi is pressing ahead with forming a government despite protests from al-Maliki against the outcome.
Iraqiya took 91 seats - two more than State of Law - in the final tally.
Allawi has said he still hopes to form a government aligned with his secular principles.
"I believe very strongly that unless we develop the concept of partnership in Iraq, no stability will occur," he said.
"The partnership, I mean it a full partnership, there is no difference between a Kurd or an Arab, a Shia or Sunni, Muslim or non-Muslim. This is only way that we move forward."
After the names of the candidates elected for the 325-seat parliament are published in daily newspapers, political blocs have three days to appeal the results.
The results will not be final until certified by the supreme court.