The dog was featured in Trump’s announcement of Baghdadi's death on Sunday, when he commended the "beautiful" and "talented" dog for its heroism, noting that no US service members were hurt or killed in the operation.
"They blasted their way in, and then all hell broke loose. It's incredible nobody was killed. Or hurt. We had nobody even hurt," Trump said. "And that's why the dog was so great."
The dog, a Belgian Malinois, was sent into a tunnel to subdue Baghdadi during the early-morning raid on a compound in Syria. However, the Daesh leader detonated a suicide vest, killing himself and three of his children and wounding the dog.
Chairman of the US Joint Chiefs of Staff General Mark Milley told reporters Monday the name of the dog remains classified to protect the identities of members of the unit it is part of.
"We're not releasing the name of the dog right now. The dog is still in theater. The dog, the canine, the military working dog, performed a tremendous service, as they all do in a variety of situations. Slightly wounded and fully recovering. But the dog is still in theater, returned to duty with its handlers," Milley said.
A senior White House official told CBS News the administration is "working on" arrangements for the dog to visit the White House, but said it might be difficult since the canine is "back at work" in the field.
Belgian Malinois are known for their work ethic and intelligence and are widely used by the military and law enforcement in the US and worldwide. A Malinois dog named Cairo famously accompanied US commandos on the 2011 raid that killed al-Qaeda leader Osama bin Laden.