As the threat from Sunni militants in western Iraq escalated last month, Prime Minister Nuri Kamal al-Maliki secretly asked the Obama administration to consider carrying out airstrikes against extremist staging areas, according to Iraqi and American officials.

But Iraq’s appeals for a military response have so far been rebuffed by the White House, which has been reluctant to open a new chapter in a conflict that President Obama has insisted was over when the United States withdrew the last of its forces from Iraq in 2011.

The swift capture of Mosul by militants aligned with the Islamic State of Iraq and Syria has underscored how the conflicts in Syria and Iraq have converged into one widening regional insurgency with fighters coursing back and forth through the porous border between the two countries. But it has also called attention to the limits the White House has imposed on the use of American power in an increasingly violent and volatile region.