Paul 'Pen' Farthing, served in Helmand in the mid-2000s and set up the charity Nowzad to look after animals in Afghanistan, with the organisation rescuing dogs, cats and donkeys.
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 plan he has dubbed Operation Ark.
Mr Farthing has previously said he will not leave the staff of his Nowzad animal charity to the "absolutely horrendous fate" they are faced with.
On Monday, a jubilant Mr Farthing announced the UK Government had granted visas for all his staff and their dependants - totalling 68 people - but the evacuation of the shelter’s animals has remained a sticking point.
Watch: Mr Farthing explained last week why he was not leaving Afghanistan without his staff.
Mr Farthing tweeted he had been "left to fend" for himself in Afghanistan after being "cut off" by his Ministry of Defence (MOD) support line.
Speaking to Sky News, Mr Wallace said: "His workforce have been offered, as entitled personnel, places, and they will be able to be called forward, but I can't guarantee in this window they will be processed onto aircraft, all I can say is they qualify."
On Tuesday, Nowzad supporters announced a privately chartered Airbus A330 – funded by donations – was on standby to fly to Kabul to rescue the group's workers and animals.
But Mr Wallace said this was still not a "magic wand", insisting the major problem with evacuations in Kabul has been getting people safely into and through the airport.
He said the chartered plane would merely "block the airfield" and "sit there empty" as the processing of the thousands of people trying to leave Kabul would be prioritised over the airlifting of animals.
However, in a series of tweets early on Wednesday, Mr Wallace said if Mr Farthing arrived at the airport with his staff and animals, officials would help them to leave on the chartered flight.
He added that if Mr Farthing and his staff chose not to bring the animals they could board an RAF flight, as all had been granted leave outside the immigration rules (LOTR) to fly to the UK.
"Now that Pen Farthing's staff have been cleared to come forward under LOTR I have authorised MOD to facilitate their processing alongside all other eligible personnel at (Kabul airport)," Mr Wallace tweeted.
"At that stage, if he arrives with his animals we will seek a slot for his plane."
Mr Farthing has previously said the animals would be transported in the aircraft’s hold and that once his staff were accommodated any spare seats on the plane could be filled by other people cleared for passage by UK authorities, with the flight able to take 250 passengers in total.
"The cargo hold is empty – we put the dogs and cats in there!! And 250 people above in the cabin!" Mr Farthing tweeted.
A Nowzad supporter in the UK, veterinary surgeon Dr Iain McGill, had said people were ready to take in the animals once they arrived in the UK.
Dr McGill said: "Pen has got transport for all the staff and animals and himself safely to the airport.
"In the airport, they can wait for a few days until the plane arrives, they just need to get safe. Pen doesn’t have that long a window because of the military operation."
Last week, Mr Farthing told Forces News the Afghanistan evacuation of Kabul Airport is a "humanitarian disaster that is just getting worse and worse every day".
Watch: The 'world is watching' British troops in Afghanistan as withdrawal deadline looms.
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Cover image: Paul 'Pen' Farthing.