BAGHDAD—Islamist extremists who have overrun swaths of Iraq made a rare retreat in an area hit by U.S. airstrikes and gave up some territory they had won from Kurdish forces, in an early sign of impact from the three-day-old American campaign.

But the country's constitutional crisis deepened as the prime minister denounced the president and security forces were deployed across the capital, raising concerns over another U.S. goal—seeing an orderly change in Iraqi leadership. After a legal deadline to nominate a new prime minister passed at midnight, incumbent Nouri al-Maliki's own party agreed to name a candidate to replace him, amid fears that he intended to prevent such a development by force or intimidation. Mr. Maliki had not agreed to step aside.

U.S. officials said they were alarmed by Mr. Maliki's fiery speech Sunday and, in particular, his call for the Iraqi army to protect the country's constitution. Obama administration officials even made calls to Iraqi politicians because of concerns Mr. Maliki was moving against his political opponents, but found none of them had been detained or placed under house arrest. An official at Iraq's Ministry of Interior said the added security was deployed in anticipation of a terrorist attack and wasn't related to political conflict.

According to a senior Shiite politician, the new choice of Mr. Maliki's State of Law party was Haider al-Abadi, who has been tipped previously as a likely candidate and recently was chosen as a deputy parliamentary speaker. U.S. officials said consensus was building around a candidate they believe would better unify the country, but declined to name him.

Meanwhile, the U.S. attacked the radical Sunni group Islamic State for a third straight day, and said the airstrikes appear to have slowed the insurgents' rapid advance over the past week toward Erbil, the capital of the semiautonomous Kurdistan region. So far, the primary focus of the strikes has been on the forces closing in on the city, where Kurdish Peshmerga fighters have been struggling to turn back the offensive. The most recent round of strikes occurred at 4 p.m. EDT on Sunday, destroying several vehicles part of an Islamic State convoy moving to attack Kuridsh Peshmerga militia forces, officials said Monday.