The attack on Wednesday, claimed by the Taliban, triggered a long-drawn gunfight near the airfield, some 50 km north of Kabul, in Parwan province, Efe news reported.
Interior Ministry spokesperson Nasrat Rahimi said on his Twitter account that two civil, including a woman, were killed and 73 civil injured.
Bagram district governor Shukoor Qudusi told EFE that the blast was so strong that its impact "destroyed and damaged several houses in nearby villages".
The governor said there were two more smaller blasts after the first major explosion near the airfield.
Provincial governor's spokesperson Wahida Shahkar said the gunfight between security forces and at least 10 Taliban fighters was "still going on".
She only confirmed the death of a woman in the explosion that occurred "close and very near" to the US airbase around 6 in the morning.
Shahkar said over 60 civil were wounded, most of them mildly.
A NATO Resolute Support mission spokesperson said, "Some service-members have been evaluated for minor injuries as a result of the initial attack".
Earlier, the spokesperson told EFE that there were no US or coalition casualties and the military base "remained secure throughout the attack".
"Enemy forces conducted an attack on Bagram airfield this morning, targeting a medical facility being constructed to help the Afghan people who live near the base," the spokesperson said, adding the future medical facility was "badly damaged".
"Some armed fighters remain inside the medical facility. The situation is isolated to the clinic building and Bagram airfield is not in danger and the defensive perimeter was never breached or compromised," said the spokesperson.
Taliban's Zabihullah Mujahid in a statement sent to EFE claimed the attack and said that "tens of" American and Afghan soldiers were killed and injured - a common exaggeration for the militant group.
The attack comes days after the Taliban and the US resumed peace talks in Doha, Qatar.
The peace negotiations were earlier abruptly suspended three months ago by US President Donald Trump, after an attack in Kabul in which an American soldier lost his life.
Since the end of the NATO combat mission in Jan. 2015, the US maintains one contingent within the framework of the new allied mission of advising Afghan troops and another for "anti-terrorist" tasks.
( With inputs from IANS )