The death of Mr. Afari would represent a serious blow to the leadership of the Islamic State. But Iraqi officials have frequently made claims about the deaths of terrorist leaders that later turned out to be untrue.

A previous leader of Al Qaeda in Iraq, the predecessor to the Islamic State, was claimed to have been killed several times, and his supposed death became a running joke in the Iraqi media. The death of another well-known Islamic State militant, an Iraqi named Shaker Waheeb, has on several occasions been reported by Iraqi officials, and each time he has emerged alive.

Mr. Afari, who also goes by several other assumed names, was recently added to a United States government list offering rewards for information leading to the death or capture of Islamic State leaders. The list, which identified Mr. Afari as Abd al-Rahman Mustafa al-Qaduli and said he had been a top deputy to Abu Musab al-Zarqawi, the former leader of Al Qaeda in Iraq, offered a $7 million bounty for Mr. Afari.

Mr. Afari is said to be the deputy to the Islamic State leader Abu Bakr al-Baghdadi, who himself has been the subject of rumors that he was wounded in an airstrike in March. On Wednesday, Gen. Tahseen Ibrahim, the spokesman for Iraq’s Ministry of Defense, said that the best intelligence he had seen suggested that Mr. Baghdadi had, indeed, been wounded, but was still in charge of the organization.