BAGHDAD -- U.S. Defense Secretary Ash Carter’s push for Iraq to let Turkey play a role in the Mosul battle encountered resistance Saturday from Iraq’s prime minister, who said his country’s forces will oust Islamic State of Iraq and Syria militants from the northern city.

“I know that the Turks want to participate, we tell them thank you, this is something the Iraqis will handle and the Iraqis will liberate Mosul and the rest of the territories,” Haider al-Abadi said through a translator after meeting with the Pentagon chief in Baghdad.

An American official told CBS News that there was some evidence of worsening morale among ISIS fighters in Mosul and that some locals have killed the men manning ISIS checkpoints.

Some ISIS fighters have wounded themselves to get off the front lines, and ISIS has conscripted members of the religious police, who are not happy about it, the official said.

Iraqi, Kurdish and other local forces will handle the battle for Mosul, al-Abadi said.

“We don’t have any problems,” he said, adding that if help is needed, “we will ask for it from Turkey or from other regional countries.”

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He acknowledged that both sides have made recommendations and that they will meet again, suggesting that the door may still be open to some compromise.

Carter, who arrived in Iraq on Saturday to meet with his commanders and assess the progress in the opening days of the Mosul operation, told reporters that the issue of a Turkish role in the military campaign is a difficult subject.

The U.S. role “is to work with our partners in the coalition and the Iraqi government to try to resolve issues like this and make sure that we’re all focused” on fighting ISIS. “I am confident that we can play a constructive role there.”

One day earlier, Carter met with Turkish leaders in Ankara and said there was “an agreement in principle” for a Turkish role. Carter stressed at the time that any final decision would be up to the Iraqis, while expressing optimism the Turks and Iraqis could settle their differences.

His visit to Iraq came two days after a U.S. service member was killed outside Mosul, underscoring the risk that American troops are taking as they advise Iraqi forces in the fight.

Carter, who already has been to Iraq twice this year, has overseen the steady increase in the number of U.S. forces deployed to the fight and the growth of America’s effort to train and advise Iraqi troops. In his two earlier visits, Carter announced White House decisions to increase the U.S. troop level there. There was no such announcement Saturday, but Carter made it clear that the U.S. stands ready to do more, if the Iraqis or his commanders identify a particular need.

The American official who spoke to CBS News said that he didn’t anticipate adding additional U.S. forces.

Some 500 Turkish troops at a base north of Mosul have been training Sunni and Kurdish fighters since last December. The Iraqi government says the troops are there without permission and has called on them to withdraw. Turkey has refused, and insists it will play a role in liberating the city.

The U.S. service member killed this week was the fourth U.S. combat death in Iraq since the U.S. began military operations against ISIS in August 2014. It was the first since the Mosul operation began.

The U.S. military said a fire at a sulfur plant in northern Iraq set by ISIS on Thursday was creating a potential breathing hazard for American forces and other troops at a base south of Mosul that’s being used by troops as a staging area.

They said some troops at the base were wearing protective masks, and that air samples were sent to the U.S. Defense Threat Reduction Agency for analysis. Officials estimated it could take two to three days to put the fire out.

U.S. defense and military officials have said that while the offensive has started well, they expect the complex fight for the city to get more difficult. They said they will be watching to see how aggressively the militants fight and whether more leaders flee the city.

“I’m encouraged by what I see so far, it is proceeding according to our plan,” Carter told reporters at the end of his day in Baghdad.

Lt. Gen. Steve Townsend, the top U.S. commander in Iraq, said enemy resistance has stiffened and has proved “significant.” But, he said, “the Iraqis expected this and they’re fighting through it.”

He said ISIS forces have used roadside bombs, multiple car bombs, snipers and “even anti-tank guided missiles” against the Iraqi troops.

Asked how long U.S. troops would remain in Iraq once Mosul is retaken, Carter said the U.S. would continue any necessary counterterrorism operations to protect the United States, and that troops could keep training and advising Iraqi forces, if requested by the Baghdad government.

The American official who spoke to CBS News said that the battle for Mosul is “a multi-month endeavor” and the military operation “will take a long time.”

“Anyone who predicts weeks, not months, hasn’t been watching ISIS very long,” the official said.

The U.S. estimates there are between 3,000 and 5,000 ISIS fighters in the Mosul area, but some leaders probably have fled. A key factor will be how long those midlevel commanders stay or whether they decide to leave.

More than 4,800 U.S. troops are in Iraq and there are more than 100 U.S. special operations forces operating with Iraqi units. Hundreds more U.S. forces are playing a support role in staging bases farther from the front lines.