KABUL, Afghanistan — Days after President Trump’s announcement of a new strategy for Afghanistan, the top American officials in Kabul said Thursday that a promised increase in United States military personnel and air power was already underway in the country.

At a news conference in the Afghan capital, the military commander for United States and NATO forces in Afghanistan, Gen. John W. Nicholson, said that the influx of new troops — mostly trainers for the Afghan security forces — would continue over the next few months.

He did not provide details on the number of troops, and he emphasized that American forces would remain in Afghanistan with no prestated timeline for withdrawal. Previous reports have suggested that the increase, which Mr. Trump has put in the hands of Defense Secretary Jim Mattis, a former Marine general, would amount to around 4,000 additional American troops.

They would be joining an American force that officially totals about 8,400, but that the Pentagon recently acknowledged to The Wall Street Journal is closer to 12,000. More than 4,000 troops from other NATO countries are also said to be in Afghanistan.