Defense Secretary Ashton Carter offered Iraqi leaders increased U.S. support and firepower Wednesday in hopes of bolstering the battle against Islamic State, which still controls some of Iraq’s largest cities.

In a meeting with Prime Minister Haider Abadi, Carter offered to send U.S. military advisers to assist Iraqi army commanders at the brigade level, rather than at headquarters, and to send Apache attack helicopters to help Iraqi troops struggling to recapture Ramadi.

The Iraqi leader, who heads a fractured government that remains deeply suspicious of U.S. intentions, declined deeper U.S. involvement for now, Carter told reporters after the meeting.

Defense Secretary Ashton Carter offered Iraqi leaders increased U.S. support and firepower Wednesday in hopes of bolstering the battle against Islamic State, which still controls some of Iraq’s largest cities. […]

“The U.S. is willing to do more … to support the Iraqi security forces,” Carter said. “Everything we do, and the coalition does, in Iraq is subject to the principal of Iraqi sovereignty and therefore his permission.”

The Obama administration is depending on Iraq’s military to carry the bulk of the battle against the militants. Coalition warplanes have backed Iraqi’s effort by launching more than 5,800 airstrikes in the country.

President Barack Obama has sent about 3,500 trainers and advisers to Iraq since mid-2014 to assist Iraqi and Kurdish forces. Carter made clear he agrees with Obama’s vow not to send thousands of U.S. ground troops back to the country.

“The ways we … can contribute to Iraqi success on the battlefield is by training Iraqi units, providing air support to Iraqi units, and possibly operating with Iraqi units and things like that. Not replace them,” he said.

In some areas, the Shiite-led government also has relied on Iranian forces, and Iraqi Shiite militias backed by Iran. U.S. officials fear their involvement could create a backlash in Sunni areas and fuel another sectarian bloodletting like the civil war that ravaged Iraq after the U.S.-led invasion in 2003.

“We have to be attentive to some of the political realities that surround us every single day,” said Lt. Gen. Sean MacFarland, who heads the U.S. military campaign in Iraq and Syria. “There are a number of complex relationships that the government of Iraq has to tend to, and we are here in Iraq at the behest of that government. So we sometimes have to adjust the things we do, like on a chessboard.”

Iraqi forces have recaptured Tikrit and pushed militants back from large parts of northern Iraq in recent months. But militants remain in control of Mosul, their self-declared capital in the country, and a U.S.-backed Iraqi ground assault aimed at retaking Ramadi has dragged on for months.

Abadi expressed optimism about the road ahead.

“I think we are on the verge of breaking the back of Daesh, I hope,” he told Carter at a Saddam Hussein-era palace where they met, using an Arabic acronym for Islamic State.

“We are trying to concentrate our position,” he said. “Of course Daesh is a terrorist organization and it has a military means at the moment, so we are fighting two fronts. We are fighting a military war and we are fighting a war against terrorism inside our cities.”

Backed by coalition airstrikes, the Iraqi army is trying to dislodge 350 to 500 militants said to be entrenched in Ramadi, about 60 miles west of Baghdad. The Iraqi army has deployed about 10,000 troops to try to retake the city.

The Iraqi forces recorded a small success Tuesday when they repulsed an Islamic State attack against the Anbar Operations Center, a government command facility that Iraqi forces had only recaptured last week.

U.S. and other coalition warplanes also launched five airstrikes against militant positions in Ramadi and its outskirts Tuesday.

But much of the city remains in the militants’ control. They have planted hundreds of booby traps to slow a government assault.

U.S. officials view the back-and-forth battle as a test of efforts to bring Iraq’s fractured security forces into a common front against the Sunni extremists.

The Pentagon has increased support in Iraq since the Nov. 13 terrorist attacks in Paris showed the threat from Islamic State was quickly expanding.

Carter deployed about 100 special-operations troops to northern Iraq to conduct raids, free hostages, gather intelligence and capture Islamic State leaders. The Pentagon also stepped up deliveries of thousands of antitank missiles, vehicles and other military hardware.

In an interview, Mowaffak Rubaie, a member of the Iraqi parliament, called for “closer consultation” between Baghdad and Washington, and “speeding the process of weapons delivery.”

Carter is making his second visit to the Iraqi capital since he was confirmed as Defense secretary in February. He is visiting several U.S. allies in the region this week to meet government leaders, military commanders and troops involved in the fight against Islamic State.