Marine Gen. Joseph Dunford, chairman of the Joint Chiefs of Staff, said last week he and Defense Secretary Ash Carter had agreed the U.S. should increase its military presence in Iraq – officially capped at 3,870 troops but with an actual tally of around 5,000.

"I brought it to the secretary, the secretary will engage with the president on what the president has asked us to do, which is to come to him with ideas that will allow us to maintain that momentum," Dunford told reporters here after change-of-command ceremonies for U.S. Central Command and U.S. Special Operations Command. "Those recommendations are being made, and the president will have an opportunity to make some decisions here in the coming weeks."

The momentum Dunford referenced now centers on the Iraqi military's offensive to retake the key city of Mosul, which saw a shaky start amid reports of desertions and disorganization in the face of an entrenched enemy. Defense officials have dismissed these reports as little more than the usual battle tempo at the beginning of a campaign.

The war against the Islamic State group, which President Barack Obama has insisted will not become another protracted U.S. ground conflict in the Middle East, adopted a new air of permanence here this week, with a top defense official suggesting the number of Americans deployed to Iraq and Syria soon may increase.