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After the Islamic State army in June swept out of Syria into northern Iraq and took the large city of Mosul and then moved south toward Baghdad, Defense Secretary Chuck Hagel had a new perspective.

“Make no mistake, and this country should not make any mistake on this, nor anyone in Congress — ISIL may not appear to be an imminent threat to the United States,” he said. “It is a threat to the United States. It is a threat, a clear threat to our partners in that area, and it is imminent.”

But two weeks ago, Deputy National Security Adviser Tony Blinken said no, the Islamic State is not a threat to Americans.

“Right now, their focus is not on attacking the U.S. homeland or attacking our interests here in the United States or abroad,” Mr. Blinken told CNN’s Jake Tapper.

White House officials attempted to portray this summer’s failed rescue mission as proof that the U.S. will do whatever it can to rescue Americans held captive abroad.

“The president could not be prouder of the U.S. forces who carried out this mission,” Ms. Monaco said. “Their effort should serve as another signal to those who would do us harm that the United States will not tolerate the abduction of our people and will spare no effort to secure the safety of our citizens and to hold their captors accountable.”

On Wednesday, a senior Pentagon official said the State Department is seeking a small boost in troop numbers in Iraq, similar to the incremental increases the Obama administration has approved for the U.S. military operation in the northern part of the country.

“There is a request by the State Department for additional U.S. personnel, and my understanding is that it’s for less than 300 [troops],” the official said.

The Obama administration has placed 849 military personnel in Iraq and tasked them with conducting airstrikes, conducting assessments and providing Iraqi forces with intelligence, Pentagon officials said.

Additionally, the framework for an international coalition to combat the Islamic State began to form Wednesday, when Germany and France said they were willing to help Kurdish fighters battle the militants.

Germany and Italy announced Wednesday that they were prepared to join Britain and France in delivering weapons and other military gear such as helmets and security vests to the Kurdish fighters.

Ben Wolfgang and Maggie Ybarra contributed to this article, which is based in part on wire service reports.