The historic Soviet-built air strip was the main base for US forces in Afghanistan following the September 11, 2001, attacks up until their 2021 withdrawal led to the takeover by the Islamist Taliban movement.
"The attack came in retaliation for Muslim prisoners in Taliban prisons, especially after their transfer to the notorious 'Bagram' prison, in a repeat of the American era and its practices against prisoners," Islamic State said in a post on its Telegram channel.
In the early years of the Afghan war under US president George W. Bush, the CIA used Bagram as a "black site" detention centre for terrorism suspects, subjecting them to abuse that US president Barack Obama would later acknowledge as torture.
The Acting Head of Prison Administration under the Taliban government, Mohammad Yusuf Mestari, said on Sunday that the prison at Bagram was being "reactivated" after it had not been in use for years.
He said the facility had been handed over the General Directorate of Intelligence and that some prisoners had been transferred there - but did not specify who they were.
Kabul police said all six killed on Monday in the suicide bombing were civilians and did not say what the target was.
Islamic State, however, claimed 45 people including Taliban members had been killed when its bomber targeted people working with judicial services.
Reuters could not independently verify the casualty figures or the target.
Afghanistan's Taliban-run interior ministry did not immediately respond to a Reuters request for comment.
Islamic State-Khurasan, a local affiliate of the Middle East-based Islamic State, has waged an insurgency against the Taliban who they see as their enemies.
Taliban authorities say they have mostly crushed the group, even as it continues to carry out attacks in Afghanistan.
Islamic State has claimed a number of attacks internationally including a stabbing rampage in Germany, a concert hall assault in Moscow and a memorial bombing in Iran.
