A US general said 12 US service members have been killed and 15 more have been injured.
Marine Corps General Kenneth McKenzie, commander of US Central Command, told a news conference: "At this time we know that 12 US service members have been killed in the attack, and 15 more service members have been injured.
"A number of Afghan civilians were also killed and injured in the attack."
Boris Johnson, who chaired a COBRA meeting following the events, said on Thursday the "overwhelming majority" of eligible people have already been helped by the RAF to flee the Taliban and "we are going to keep going up until the last moment" as the 31 August deadline rapidly approaches.
He added: "I want to stress that we're going to continue with that operation [Pitting], and we're now coming towards the end of it, the very end of it, in any event, and we've already extracted the overwhelming majority of those under both the schemes – the eligible persons, the UK persons, UK nationals plus the Afghans, the interpreters and others.
"It's been a totally phenomenal effort by the UK, there's been nothing like it for decades and decades.
"Clearly what this attack shows is the importance of continuing that work in as fast and as efficient a manner as possible in the hours that remain to us and that’s what we're going to do."
Pentagon spokesman John Kirby confirmed two blasts occurred in a "complex attack" outside Hamid Karzai International Airport, killing "a number of US service members" and injuring others, as "a number of Afghans fell victim to this heinous attack".
The Ministry of Defence said at an early stage that there had been no UK military or Government casualties reported.
The explosions came following warnings that a terror attack could be launched in the final phase of the evacuation.
On Thursday morning, James Heappey, the UK's Armed Forces Minister, warned of an "imminent and lethal" threat of an ISIS attack targeting Kabul airport.
He had urged people queuing outside Hamid Karzai International Airport to move to safety amid concerns over an affiliate of the so-called Islamic State (IS) in Afghanistan, known as ISIS-K.
The MOD did not confirm or deny whether ISIS-K was believed to be behind the explosions.