A sluggish State Department response to the Taliban’s rapid takeover of Kabul, Afghanistan’s capital, has stranded thousands of Afghans who helped the United States and are now clamoring to be evacuated as they wait for their immigration visas to be approved, two U.S. officials said.

As many as 6,000 people — including former interpreters and cultural and political advisers — were on standby to be flown out of Kabul’s airport late on Thursday night or early Friday alone, after a days-long pause in the processing of visas for Afghans who had worked for the American military or embassy during the 20-year war, the State Department said.

Thousands more are expected to be vetted and evacuated on a daily basis after a small influx of consular officers and other diplomats — including the former ambassador to Afghanistan, John R. Bass — arrived in Kabul on Thursday to speed the visa processing. Diplomats are also deploying to Qatar and Kuwait, where U.S. military bases will serve as way stations for people arriving from Afghanistan as they search for a final destination.