BAGHDAD — Iraqi forces on Tuesday reported progress in the military operation to retake the city of Ramadi from the Islamic State group, saying they made the most significant incursion into the city since it fell to the militants in May.

Losing Ramadi — the capital of sprawling western Anbar province and Iraq’s Sunni heartland — was a major blow to the Iraqi government. It was the government’s biggest defeat since IS militants swept through areas in the country’s north and west, including Iraq’s second-largest city of Mosul, in the summer of 2014.

Iraqi forces announced a counteroffensive shortly afterward Mosul fell but progress has been sluggish and clawing territory back from IS has proven more difficult than expected.

Col. Steve Warren, a spokesman for the U.S. military in Baghdad, said there are 250 to 350 Islamic State fighters in Ramadi, as well as several hundred outside the city on the northern and western perimeter.

“I think the fall of Ramadi is inevitable,” Warren told Pentagon reporters. “But that said, it’s going to be a tough fight … it’s gonna take some time.”

He said American military advisers remained outside the city at al-Taqaddum, a desert air base that is serving as a training site. It was a U.S. military hub during the 2003-2011 war.

Iraqi spokesman Sabah al-Numan said troops crossed the Euphrates River north of the city and its Warar tributary to the west and pushed into downtown Ramadi.

From the south, troops led by the counter-terrorism agency made progress in the Dubbat and Aramil neighborhoods, about 3 kilometers (less than 2 miles) from the city center, Gen. Ismail al-Mahallawi, the head of operations in Anbar province, told AP.

Sporadic clashes broke out and advancing Iraqi forces were forced to remove roadside bombs planted by the extremists, al-Numan added.

On Tuesday, the Dubbat neighborhood saw heavy fighting, with one soldier killed and 14 wounded, said an official in the Anbar operations room, speaking on condition of anonymity because he was not authorized to brief the media.