Citing two Afghan officials, The New York Times said that the shootout had caused the death of five or six American soldiers and six Afghan soldiers.
Military Times also learned from an anonymous source that at least two members of the 7th Special Forces Group were killed and several others wounded.
The firefight took place in Nangarhar Province, where both the Afghan Taliban and the Islamic State branch in Afghanistan have a presence.
It was not clear whether the firefight resulted from Taliban infiltrators or an "insider attack."
Sonny Leggett, a spokesman for U.S. forces in Afghanistan, said in a statement that "we are assessing the situation and will provide further updates as they become available."
U.S. President Donald Trump has long grumbled about the U.S. military presence in Afghanistan, which started in 2001, calling it "ridiculous."
Earlier this week, Trump said in his State of the Union address that his administration is working to end the war in Afghanistan.
Speaking to a joint session of Congress in the House of Representatives, Trump noted that the U.S. military is making progress in Afghanistan, and peace talks with the Taliban militant group are underway.
Peace talks between the United States and the Taliban began last year but have been interrupted at least twice after Taliban attacks on U.S. military personnel in September and December.
The United States maintains roughly 13,000 troops in Afghanistan, which mostly provide training missions to local Afghan forces while conducting counterterrorism operations.
The war in Afghanistan is the longest one in U.S. history. The death toll of U.S. service members has surpassed 2,400 since the country invaded Afghanistan in 2001.