The 65-year-old Allawi held the post of Communications Minister in the government of Nouri al-Maliki before quitting in 2012 over what he described as "political interference" by the then-Prime Minister.
The attackers entered the area in a convoy of four-wheel drive vehicles and shot at those gathered for the demo.
Nechirvan Barzani is the second person to hold the office of president, occupied by his uncle from its creation in 2005 until late 2017 when he quit following an independence referendum hotly opposed by Baghdad.
The US last year shut its consulate in the protest-hit southern Iraqi city of Basra, blaming "indirect fire" by Iran-backed forces and warning its rival of retaliation for any damage.
