The strikes took place 15 km (9 miles) away from the border, said a statement from the PMU.
It said one fighter was killed and another seriously wounded, Reuters reported.
An earlier statement from the PMU’s Anbar operations command said two fighters were killed.
The PMU said two unmanned aircraft had carried out the attack. It accused the United States of providing air support to Israel for the strikes.
A security source said there were two airstrikes, one of which struck the headquarters of a local brigade, while the other struck a convoy of cars leaving the building.
The attack, which took place in the afternoon, happened after a series of explosions in recent weeks at weapons depots belonging to the popular group.
Israeli Prime Minister Benjamin Netanyahu hinted on Thursday of possible Israeli involvement in the recent destruction of PMU weapons depots in Iraq.