BAGHDAD — Only a third of Fallujah has been ‘‘cleared’’ of Islamic State militants, the US-led coalition said Tuesday, days after the Iraqi government declared victory in the city west of Baghdad, which was held by the extremists for more than two years.
Other parts of the city are ‘‘contested,’’ said US Army Colonel Christopher Garver, the Baghdad-based spokesman for the coalition, with clashes underway between Iraqi forces and Islamic State fighters. Most of the cleared terrain is in the south of the city and ‘‘clearing operations continue outward from the city center,’’ Garver said.
Iraqi forces pushed into the center of Fallujah on Friday, retaking a government complex and the central hospital. That evening Brigadier General Haider al-Obeidi, with Iraq’s special forces, said his troops controlled 80 percent of the city.
Iraqi Prime Minister Haider al-Abadi said Friday that Fallujah had ‘‘returned to the embrace of the nation,’’ and that remaining Islamic State pockets would be ‘‘cleaned out within hours.’’
But in recent days there have been persistent clashes between Iraqi forces and Islamic State fighters holed up in dense residential neighborhoods along the city’s northern edge.
‘‘What it looks like is [an Islamic State] defensive belt around the city with not as stiff defenses inside,’’ Garver said, explaining that as Iraqi forces move out from the city center they may encounter additional pockets of stiff resistance.
‘‘That could be their toughest fighting,’’ Garver added.
Iraqi commanders on the ground say their forces continue to make progress and have killed hundreds of militants.
Iraqi special forces backed by US-led airstrikes have taken control of the neighborhoods of al-Shurta and al-Jughaifi, Obeidi said on Tuesday. He said Iraqi military engineers were clearing the streets and buildings of leftover bombs.
The top special forces commander for the Fallujah operation told local al-Sumaria TV late Monday that the offensive killed 2,500 Islamic State fighters. Lieutenant General Abdul-Wahab al-Saadi offered no evidence to back up his claim. Iraqi troops have not disclosed their losses in Fallujah, though the Islamic State group claims to have killed dozens.
The operation has fueled an exodus of thousands of families, overwhelming camps for the displaced run by the government and aid groups.
The United Nations refugee agency said Tuesday that more than 85,000 people have fled Fallujah and the surrounding area since the offensive began. Ariane Rummery, spokeswoman for the United Nations High Commissioner for Refugees, said she expected that thousands more ‘‘could still be planning to leave the city’’ an appealed for more aid donations.
The US State Department announced an additional $20 million in aid to the UNHCR in response to appeals for emergency needs in Fallujah.
‘‘This will be part of a larger package of humanitarian assistance that will be announced later this year, so there’s more aid coming,’’ State Department spokesman John Kirby said.
Since 2014, the United States has sent more than $778 million in humanitarian aid to Iraq, according to the State Department. Since August 2014, it has spent a total of $7.5 billion on operations against the Islamic State in both Iraq and Syria, according to the Pentagon.
Fallujah is the last Islamic State bastion in the sprawling Anbar province. The Islamic State still controls pockets of territory in Iraq’s north and west, including Mosul, the country’s second-largest city.
More than 3.4 million Iraqis have fled their homes since the Islamic State swept across northern and western Iraq in the summer of 2014, according to UN figures. More than 40 percent of the displaced are from Anbar province.
A series of attacks in and around Baghdad, including one targeting anti-Islamic State Sunni fighters, killed at least 12 people and wounded 42 others on Tuesday, police and health officials said on condition of anonymity as they were not authorized to release information.
No group has claimed responsibility for the attacks, but they bore the hallmarks of the Islamic State. The near-daily attacks are seen by Iraqi officials as an attempt to divert security forces’ attention from the front lines.