Signs of strain have emerged recently between the United States and Iraq over the timetable and military components of a campaign to retake major population centers occupied by the Islamic State.

Defense Secretary Ashton B. Carter told Congress this week that the U.S. Central Command was “inaccurate” when it told reporters recently that an offensive in Mosul could begin as early as April. But that timeline had already provoked a retort from Carter’s Iraqi counterpart, who said the United States was “not familiar” with Iraq’s battle plan for the northwestern city.

Speaking at a news conference late last month, Iraqi Defense Minister Khaled al-Obeidi said that Baghdad would determine the timing for the Mosul offensive.

Other officials here cite a raft of challenges that could delay or complicate offensives in Mosul and elsewhere, including the capacity of Iraqi government forces to hold recaptured territory and ongoing problems coordinating with Kurdish forces, called the peshmerga.

Perhaps most worrying are growing sensitivities about the role played by Iranian-backed Shiite militias in government offensives being carried out in mainly Sunni areas. A key test is the campaign to retake the majority-Sunni city of Tikrit, 125 miles south of Mosul, from the Islamic State.

There are concerns that that battle will turn into a bloody war of attrition. Just a fraction of at least 10,000 pro-government fighters massing around Tikrit are from the official armed forces, highlighting the military’s reliance on militias and volunteers as the United States and other coalition partners seek to rebuild and retrain Iraq’s security forces.

[View: The military assault on Tikrit, Iraq, in pictures]

Secretary of State John F. Kerry acknowledged Wednesday that Maj. Gen. Qasem Soleimani, head of the Iranian Revolutionary Guard Corps’ elite Quds Force, was “on the ground” and “playing a role” in the Tikrit operation, although he insisted that the Iraqi government was in charge and that the need for the militias was only temporary.

“As the Iraqi army stands up more and more, militias and external actors are going to be less and less imperative and needed,” Kerry said, speaking at a news conference in Riyadh on Thursday with Saudi Foreign Minister Prince Saud al-Faisal. “But that’s not where they are,” he said of the Iraqi army forces in Tikrit.

For now, Kerry said of Iranian and militia participation, “we take it the way it is.”

Faisal, whose Sunni government is training Iraqi military forces along with the United States, appeared far more concerned. “The situation in Tikrit is a prime example of what we’re worried about,” he said at Kerry’s side. “Iran is taking over the country.”

Kurdish intelligence chief Masrour Barzani echoed the Saudi concerns in an interview last week in Irbil, the Kurdish capital. “Who is going to do the job?” he said of offensives in Tikrit and Mosul. “Without the Iraqi army and, more specifically, Sunni elements within these forces, it will not produce the results that we all hope for.”

U.S. Army Gen. Martin E. Dempsey, chairman of the Joint Chiefs of Staff, told Congress this week that local Sunni leaders are supportive of the Tikrit operation, even with Shiite militia participation. But Dempsey, who testified along with Carter, said: “We’re watching carefully. If this becomes an excuse to ethnic cleanse, then our campaign has a problem, and we’re going to have to make a campaign adjustment.”

Tikrit, now largely devoid of civilians, is seen as far less problematic than Mosul, where the large remaining civilian population has prompted questions not only about who will lead that offensive but what force will secure the territory.

Several hundred Sunni tribesmen are among the pro-government ranks in Tikrit, but whether they, along with local police forces, are capable of holding regained ground there or in a future Mosul operation remains a major concern, according to Maj. Gen. Fadhil Jalil al-Barwari, head of Iraq’s special forces. The Shiite militias have been accused of mass killings of Sunnis, and that, along with long-standing resentment of the Shiite-dominated army forces that served in Mosul in advance of the Islamic State’s takeover, could turn the local population against any liberating force.

“I can attack, but I need someone to hold the ground,” said Barwari, whose elite U.S.-trained forces have led numerous operations against the Islamic State over the past year. “There is a major effort now to try to work out who will.”

Barwari, speaking in Baghdad last week, said 10 of his units are currently tied up holding territory in Ramadi, capital of the western province of Anbar. The province abuts Baghdad, making it a strategic worry.

Local provincial councils and tribal leaders have been trying to muster fighters capable of holding territory from which the Islamic State has been ousted, but initiatives, including a training camp for exiled Mosul police forces, remain uncoordinated and ill-equipped.

“I don’t think that they are quite ready yet,” Barzani said. “A combat-hardened force requires more than just training.”

Kurdish officials, meanwhile, say they face political limits on how far their forces can push outside Kurdish areas and note that they still lack the heavy weaponry they need to provide support in any major operation.

“Nothing heavy has been delivered, neither from the United States or other allies,” Barzani said, adding that some Kurds suspected a political decision not to arm them heavily because of fears of future secessionist aims. “We are frustrated. We are disappointed. We believe the peshmerga are risking their lives, and we’d hope that their lives were worth more than equipment.”

A Defense Ministry official here said last week that “there is no timeline” for the Mosul offensive. “Some people are talking about June, but it depends on the success of the Iraqi forces. It’s a huge challenge, and there are questions about whether the militias can participate,” said the official, who spoke on the condition of anonymity because he was not authorized to speak to the media.

The United States has estimated that just 1,000 to 2,000 Islamic State fighters are in Mosul, but Iraqi officials put the number much higher. According to Barzani, Kurdish intelligence points to thousands of Islamic State reinforcements arriving in recent months, many from Syria.

Barwari said that Iraqi communications intercepts have shown that the Islamic State is trying to expand its presence in Mosul. He said that there were 3,500 Islamic State fighters in the city in December, before the extremist group cut cellphone and Internet networks there, making intelligence more difficult to obtain.

Iraqi officials and military leaders also have questioned whether retaking Mosul should be a priority now or should wait until the Islamic State’s supply lines in western Iraq can be cut.

“Before Mosul, we need to do something in Ramadi and break the siege on Haditha” in Anbar province, said Karim al-Nouri, spokesman and military commander for the Badr Organization, which has emerged as one of the most powerful Iranian-backed paramilitary groups in the country.

DeYoung reported from Washington. Mustafa Salim in Baghdad contributed to this report.

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