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Paul Farthing, known as Pen, who founded the Nowzad shelter in Kabul, is hoping to get his animals out of the country alongside his animal shelter staff.
Farthing, who was outside the airport in a car when the incident occurred said: "We’re fine but everything is chaos here at the moment.
"All of a sudden we heard gunshots and our vehicle was targeted, had our driver not turned around he would have been shot in the head by a man with an AK-47.
"We’ve been in the airport, and back out of the airport; the whole thing’s a mess.
"There’s not much more I can say at the moment, I need to make sure the animals and everyone is safe."
The gunfire appears to have happened alongside a pair of targetted explosions - which are believed to have been caused by suicide bombers - near Kabul airport.
Both explosions took place near the Abbey Gate entrance to the airport, where large numbers of refugees have gathered in recent days trying to flee the Taliban.
Watch: Afghanistan: Defence secretary slams claims ex-marine's Kabul animal rescue flight was blocked as 'total myth'
Pentagon press secretary John Kirby said: "We can confirm that the explosion near the Abbey Gate of the Kabul airport has resulted in an unknown number of casualties."
Witnesses on the ground said they believe several people have been killed.
Thousands of people have thronged around the airport in recent days as they try to board evacuation flights out of Afghanistan.
A local witness said he heard the explosion by Kabul airport as he was walking to evening prayer.
Ahmad, whose name has been changed for security reasons, said he is safe as he was some distance from the blast.
“First was explosion, and then firing started, I mean… heavy gunfire,” he said.
“But even far away people were running… there’s alarm.”
Ahmad has been hoping to flee Afghanistan but said there has been “no way” to access the airport in recent days and “tens of thousands of people” outside it.
There have been “no reported UK military or UK Government casualties” following the explosions in Kabul, the Ministry of Defence has said.
Fathering and his animals have been a cause of controversy in the UK in recent days, with many people questioning if the places on evacuation planes taken by his animals would be better used on refugees.
He has attempted several times to get into the airport but has been denied passage each time.
Defence secretary Ben Wallace denied on Thursday that the UK has blocked the animals from leaving.
Ben Wallace insisted that 'no one... had blocked a flight', before describing the claim as a 'total myth'.
The defence secretary went on to indirectly accuse Farthing's group of 'bullying, falsehoods and threatening behaviour towards MoD personnel' as the diplomatic row continued.
Since the collapse of the Afghan government, he has campaigned to have his staff and their families as well as 140 dogs and 60 cats evacuated from the country in a privately chartered Airbus A330.
Armed forces minister James Heappey warned on Thursday morning that there is “very credible reporting” of an “imminent” and “severe” threat to Kabul airport.
US officials strongly believe the ISIS-Khorasan group was behind the attack on Thursday at Kabul’s airport, a source familiar with congressional briefings on Afghanistan said on Thursday.
A second US government source familiar with intelligence activities said that while the US government is still investigating, the airport attack has “all the hallmarks” of an ISIS-K attack.
Watch: '‘She died right in my arms’: Afghan witness says small child was victim of Kabul blasts