A Kabul defence ministry official, who declined to be named, could not say exactly when the US soldier had been hurt, but it was in the last 24 hours in the Zabul province.
Clashes had resumed in the rugged area on Wednesday after a lull overnight, said the official.
About 450 Afghan government soldiers and several dozen US special forces have been chasing up to 600 Taliban fighters since Monday.
Afghan officials say up to 70 Taliban fighters have been killed, although the US military has confirmed only 14 deaths in the operation.
Most of the Taliban losses occurred during raids by aircraft of the US-led force in Afghanistan.
US-led forces toppled the ruling Taliban in 2001, following the September 11 attacks.
The last few weeks have been some of the bloodiest since the Taliban’s fall, with attacks on civilians, government troops and Afghan aide workers blamed on the militia which opposes the presence of foreign troops in the country.