KHAZIR, Iraq — Iraqi Kurdish forces advancing toward the northern city of Mosul paused Tuesday on the second day of a long-awaited offensive after the Islamic State mounted tough resistance in villages east of the strategic city.

The front lines were largely quiet after fierce fighting as Kurdish forces pushed to retake a string of villages on the edge of their territory and turn them into new footholds closer to Mosul. The city is the last main stronghold for the Islamic State in Iraq and represents a critical showdown for both sides.

However, the Iraqi military made some gains. The army’s 9th Division broke into the district of Hamdaniya, also on the eastern front, closing in on the Christian town of Qaraqosh.

The intense fight for hamlets some 30 miles from the city suggested that the militants could fight for every inch outside the city, but also that they are unlikely to be able to hold on in the face of an advance and air support from a U.S.-led coalition.

Residents who have recently fled the area and Iraqi officials with contacts inside Mosul say the Islamic State has been digging in for a fight, erecting concrete barricades and filling trenches full of oil that can be set on fire to slow advancing forces. The Islamic State claimed to have carried out 12 suicide attacks on the first day of the offensive.

The struggle for Mosul – which involves U.S. air power – is the largest and most complex so far in the battle against Islamic State militants.

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