The fighting included the shooting down of a Special Forces helicopter with 16 soldiers aboard earlier this week, US officials said on Friday.
A purported Taliban spokesman has claimed to have captured one of the soldiers.
The developments further worsened the already stinging blow the US military suffered from the deaths of the 16 aboard the MH-47 Chinook helicopter, and comes as it scrambles to deal with an insurgency that threatens three years of progress towards peace.
US military spokesman Lt. Col. Jerry O'Hara said US forces were using "every available asset" to search for the missing troops.
"Until we find our guys, they are still listed as unaccounted for and everything we got in that area is oriented on finding the missing men," the spokesman said. He declined to identify the soldiers or to say how many were missing.
The missing troops are a small team from the special operations forces, said military officials, speaking on Friday on condition of anonymity, because rescue operations were still under way.
Though the team has been missing since Tuesday, the military had refrained from discussing their situation to prevent the Taliban from setting out in search of them.
The downed helicopter had gone into the mountains near the town of Asadabad, close to the Pakistani border, on Tuesday to "extract the soldiers," O'Hara said. The team on the ground has been missing since the chopper was downed.
The Taliban claim of kidnapping a soldier came from its purported spokesman, Mullah Latif Hakimi.
Lt. Gen. James Conway : No troops
are classified as officially missing
"One high-ranking American has been captured in fighting in the same area as the helicopter went down," he told The Associated Press.
"I won't give you any more details now."
Reacting to the claim, O'Hara said : "We have no proof or evidence indicating anything other than the soldiers are missing."
Hakimi, who also claimed that the fighters shot down the helicopter, often calls news organisations to take responsibility for attacks, often with information that proves exaggerated or untrue, reports AP. His exact tie to the Taliban leadership is not clear.
At the Pentagon, Lt. Gen. James Conway, director of operations for the Joint Chiefs of Staff, said the military did not yet have a full account of all ground troops involved in the operation, but he said no troops had been classified as officially missing.
Rescuers - struggling against stormy weather, Taliban fighters and a rugged terrain - have recovered the remains of the 16 and on Friday were trying to identify them, the military said.
Conway said it appeared an unguided rocket-propelled grenade had hit the chopper. He called it "a pretty lucky shot against a helicopter."
Only eight months ago, Afghan and US officials were hailing a presidential election as a sign that the Taliban rebellion was finished.
Unprecedented fighting has left
up to 477 Taliban fighters dead
But remnants of the former Taliban regime have stepped up attacks, and there are disturbing signs that foreign fighters - including some linked to al-Qaida - might be making a new push to sow an Iraq-style insurgency.
Afghan officials say the fighters have used the porous border with Pakistan to enter the country, and officials have called on the Pakistani government to do more to stop them.
The loss of the helicopter follows three months of unprecedented fighting that has killed about 477 suspected fighters, 47 Afghan police and soldiers, 134 civilians and 45 US soldiers, including the 16 killed in Tuesday's crash.
The crash was the second of a Chinook helicopter in Afghanistan this year. On 6 April, 15 US service members and three American civilians were killed when their chopper went down in a sandstorm while returning to the main US base at Bagram.