"IS has come from three directions; we are almost besieged," Aref al-Janabi told AFP by telephone.
"So far we are still standing," he said. "We have some support from tribal fighters, but if Amriyat falls, the battle will move to the gates of Baghdad and Karbala."
Amriyat al-Fallujah lies around 35 kilometers west of Baghdad's limits, and IS fighters would have to capture a significant stretch of government-controlled land before reaching the capital.
Government forces have suffered a string of bruising military setbacks in Anbar in recent weeks, prompting some officials to warn that the entire province could fall within days.
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Soldiers pulled out of a base near the city of Heet and regrouped in a large desert airbase, while government forces struggled to hold their ground in the provincial capital Ramadi.
Some officials in Anbar have argued that anything short of an intervention by US ground forces would lead to Anbar falling into jihadist hands.
But the head of Iraq's Shiite-dominated government, Prime Minister Haidar al-Abadi, has ruled out any foreign ground intervention.
While IS fighters have not moved closer to Baghdad in recent weeks, the group has claimed responsibility for a string of deadly suicide attacks in the capital.
A suicide car bomb blast claimed by IS today killed a member of parliament who was also a prominent leader in the Iran-backed Badr Shiite militia.