U.S. forces Monday drew near Najaf's revered Shiite shrine, engaging in fierce battles with followers of radical cleric Muqtada al-Sadr as the military stepped up pressure on the insurgents to hand over the holy site to religious authorities.

Al-Sadr followers claimed that U.S. airstrikes overnight damaged an outer wall of the Imam Ali Shrine compound, which remains in control of al-Sadr's Mahdi Army militia. The U.S. military said the strikes targeted militants south of the shrine who fired on U.S. troops, and that the wall was not hit. There was no independent confirmation of the damage.

Gunfire rang out in the city throughout the day, and black smoke rose over the Old City, a neighborhood of small, twisting alleyways where fighters...