As the war against the Islamic State in Iraq shows signs of fragile progress, U.S. military leaders there are preparing to ask President Obama for the one thing he is resisting above all else in his final months — the deployment of hundreds more U.S. troops.

Military leaders directing operations against the terrorists in Iraq are readying requests for more troops and equipment they feel are needed to solidify and quicken progress toward defeating the Islamic State. These proposals have not yet been formally submitted to the White House for approval, and would first be vetted by the Pentagon leadership, but key generals have already told many in Washington they need hundreds more U.S. personnel to do the job right.

According to several senior military, congressional and administration officials, the generals on the ground, including Army Lt. Gen. Sean MacFarland, commander of coalition forces in Iraq, have been frustrated by what they see as arbitrary caps on troop levels set by the White House and a process that discourages them from directly asking for what they need. That may affect the numbers in the requests that are sent to Washington.

“What MacFarland is trying to do is again make another change,” said retired Gen. Jack Keane, former vice chief of staff of the Army. “When MacFarland is talking about increasing the numbers again dramatically, that’s what he needs to do.”

After the White House approved 200 more troops in Iraq in April, MacFarland told reporters the generals would “ask for more if we need to.” Keane said MacFarland and other generals on the ground need more tactical air controllers and advisers embedded with units closer to the fight. But they haven’t formally requested hundreds more troops due to pressure from their superiors, he said.

“They know what not to ask for because it’s not going to happen,” Keane said. “It gets cut off because the White House is not going to approve it. The level of frustration over this has really been quite extraordinary.”

The White House has incrementally raised its cap on U.S. troops in Iraq several times since the war on the Islamic State began. Officially, there are just under 4,100 U.S. troops deployed there. Military and congressional officials said about 900 more U.S. soldiers in Iraq are not counted in official numbers because they are Special Operations forces, who are often not acknowledged, or they are deployed on a temporary duty status as a way to fudge the statistics.

Gen. Daniel B. Allyn , vice chief of staff of the Army, told me the White House’s caps on troop numbers in Iraq have forced the Army to break up units and supplement deployed forces with contractors, which harms Army operations and adds costs. The number of U.S. contractors supporting the military in Iraq is unknown but estimated to be in the thousands.

“When you look at our deployment of our advise and assist formations [in Iraq], they are very leader heavy . . . what we leave back is the rest of that formation, which often limits what we can do with that force,” Allyn said. “A continued requirement to deploy formations without all their organic capability and outsourcing that through contractors has significant downside risks for the Army.”

Allyn traveled to Baghdad last month and met with MacFarland and the other generals on the ground. The generals are right now preparing detailed requests for what they believe would be necessary if the fight in Fallujah succeeds and the United States must help the Iraqi forces surround and eventually take Mosul.

“General MacFarland’s sensing, when I was with him, of what he needed, was in the works,” Allyn said. “They have a good feel for what they need and the sequencing of what they need.”

Generals involved in the mission have also been telling lawmakers and congressional staffers behind closed doors that they need more troops sooner rather than later, given the fragility of the current process and the time lag between requests and when the troops are able to contribute to the fight.

“Can the generals request an increase to the troop levels? Yes,” House Armed Services Committee Chairman Mac Thornberry (R-Tex.) told me. “But before that, the feelers go back to the National Security Council and they find out what the reaction is going to be. They understand the White House wants to keep the numbers to a minimum.”

“We’ve been slowly adding troop strength, it’s like pulling teeth to get more resources. It’s frustrating to people who are engaged in the battle,” said Sen. Lindsey Graham (R-S.C.). Graham said that U.S. military planners in Iraq have told him they have insufficient troops and resources to accomplish their mission.

Defense Secretary Ashton B. Carter and Joint Chiefs of Staff Chairman Gen. Joseph F. Dunford Jr. are said to be sympathetic to requests for more resources in Iraq but also wary of pushing the White House beyond what it is comfortable with.

One senior administration official who was not authorized to speak on the record told me that there is a sense of urgency in prosecuting the war that comes directly from the president, but that large increases of U.S. troops in Iraq carry political risk for the Iraqi government.

“There’s a certain amount of support that we can provide to the Iraqis that is essentially politically sustainable for them,” the official said. “That limits the type of support we can have in some situations.”

A spokesman for the military operations command in Iraq offered no comment in response to my request.

Senior administration officials maintain that the current campaign plan is working and that all formal requests that reach the president receive serious consideration. They also say the president insists that the fight against the Islamic State be resourced in a way that’s sustainable given the expected length of the mission and America’s other military commitments.

“More of everything is not a strategy,” said Derek Chollet, a former Obama administration White House and Pentagon official. “In every single military campaign in history, the military has wanted more. They are trying to do a job. It’s the natural impulse.”

Retired Lt. Gen. James Dubik, who was commanding general of the Multi-National Security Transition Command-Iraq during the surge in 2007 and 2008, said that even if the military gets the extra troops it wants, the Islamic State will never be defeated until the United States commits to larger diplomatic and political engagement as well, to shore up the Iraqi state and provide for long-term stability.

“We’ve been giving military assistance, which is necessary but insufficient,” he said. “We want to defeat ISIS, but merely running them out of Iraq will not defeat ISIS. They will be back.”

To critics of the Obama administration’s Iraq policy, White House reluctance to approve more troops and resources to fight the Islamic State is another example of the president’s determination to build a legacy around ending two wars started by his predecessor and removing the United States from bloody Middle Eastern conflicts.

“He doesn’t want his legacy to be that he went back into Iraq,” said Graham. “The next president is going to have to finish the job and is going to have a mess on their hands.”

If Obama decides not to fulfill the generals’ wish for more resources in Iraq, he may preserve his personal legacy by keeping troop levels low. But that comes at the cost of the Iraqis and U.S. security, and it will only pass on those tough calls to whoever succeeds him.