State television identified the man as Abu Huthaifa al-Yamani. It did not say when the strike took place or give further details. It was not immediately possible to confirm the death or whether Yamani was an aide to Baghdadi.
Iraqi security officials have not confirmed the death.
The Pentagon said on Monday it could not corroborate reports that Baghdadi had been either killed or wounded in Iraq, acknowledging conflicting media reports from the region.
"We do not have any information to corroborate reports out of Iraq that Baghdadi has been either killed or wounded," Colonel Steve Warren, a Pentagon spokesman, told reporters.
The contradictory accounts from Iraq followed U.S. air strikes on Friday night.
Islamic State, which swept through northern Iraq in June virtually unopposed by the Iraqi army, has declared a caliphate in the parts of Iraq and Syria it controls.
Falluja is an Islamic State stronghold to the west of Baghdad in the Sunni Muslim heartland Anbar Province.
The United States and its allies launched a barrage of attacks against Islamic State over the weekend, conducting 23 air strikes in Syria and 18 in Iraq against the militant group since Friday, U.S. Central Command said.
In Iraq, seven strikes hit near Baiji and others in or near Falluja, Mosul, al-Qaim, Haditha, Ramadi and Rutba.
