ERBIL, Iraq — Iraqi forces on Sunday began penetrating the narrow streets and warrens of Mosul’s heavily populated old city, in the last phase of a monthslong battle against the Islamic State militants that American commanders have described as one of the toughest in urban warfare since World War II.

The assault began at dawn, with airstrikes and a push by Iraq’s counterterrorism forces into the neighborhoods of the old city. It was met with fierce resistance by Islamic State fighters, according to commanders, suggesting that the battle, the most vicious phase in the long fight for Mosul, could go on for days or weeks.

As the fight began, security forces broadcast a message to residents of the old city over loudspeakers: “The security forces are advancing toward you and the salvation hour is near, and the enemy is losing its positions one after another, and their end is near.”

The message continued, asking civilians to flee toward security forces: “Come toward your brothers, your armed forces, and you will find the proper care.”