According to the officials, the attacker was riding a motorcycle and did not make it past any checkpoints.
Bagram District Governor Abdul Shukoor Qudosi told CNN the attacker detonated his explosives at the main entrance of the base, targeting truck suppliers.
It was not clear if the wounded US citizen was a military servicemember or a contractor.
"The explosion resulted in a small number of casualties. The casualties are being treated at Bagram medical facilities," US Forces Afghanistan said in a statement. The statement also said: "Bagram Airfield is secure and the incident is being investigated."
Taliban spokesman Zabiullah Mujahid said the group claimed responsibility for the attack in a text message to media, saying it was carried out as revenge for leaflets distributed by US-led NATO forces in the area on Tuesday.
US forces apologized for dropping a leaflet in northern Afghanistan Tuesday depicting a dog with a Koranic verse across its body -- a highly offensive image to Muslims.
The leaflet was a call to Afghans to help security forces fight terrorist groups operating in that part of the country.
"On September 5, US forces conducted a leaflet drop in Parwan Province. The design of the leaflets mistakenly contained an image highly offensive to both Muslims and the religion of Islam," Maj. Gen. James Linder, who heads US and NATO special forces in Afghanistan, said in a written statement. "I sincerely apologize. We have the deepest respect for Islam and our Muslim partners worldwide."
"There is no excuse for this mistake," Linder also said in the statement. "I am reviewing our procedures to determine the cause of this incident and to hold the responsible party accountable. Furthermore, I will make appropriate changes so this never happens again."
The attack comes just weeks after President Donald Trump announced that the US would expand its presence and remain in Afghanistan for the foreseeable future to prevent terrorists from taking advantage of a security vacuum in the country.
"The consequences of a rapid exit are both predictable and unacceptable," Trump said during a nationally televised speech last month at Joint Base Myer-Henderson Hall in Virginia.
Last week, the Pentagon announced
there were 2,600 more US troops in Afghanistan than previously disclosed publicly due to a change in its accounting procedures, bringing the total up to approximately 11,000 troops from the previously announced number of 8,400.
While US and coalition casualties have fallen in recent years since the Afghan government assumed responsibility for combat operations in 2014, there have still been incidents that resulted in US fatalities or injuries.
Last month, an attack on a NATO mission convoy in Kandahar resulted in the deaths of two US service members
. And in June, seven US service members were wounded in an insider attack at Camp Shaheen in northern Afghanistan.
Earlier in June, three US soldiers were killed, and another wounded during a joint US-Afghan military operation in Afghanistan's Achin district. The Taliban claimed responsibility for that attack.
As they have taken the lead for combat operations in their country, Afghan soldiers have also lost many of their own in combat operations.
Dozens of Afghan soldiers were killed last month during a Taliban attack on an army camp in Kandahar province.
US forces in Afghanistan serve under two primary missions -- the most being assigned to train and advise Afghan security forces along with approximately 6,000 troops from other NATO countries involved in the mission.
The remainder of those US forces carry out counterterrorism missions against al Qaeda and ISIS affiliated groups operating inside Afghanistan and other militants.
US military bases and facility throughout the world are also on high alert with the upcoming anniversary of the September 11 attacks.