The mum to the kittens has been named Morgan after midfielder and England international Morgan Gibbs-White, while the female kittens have been christened in honour of Danilo, Murillo and Ibrahim Sangare, and the sole male named after New Zealand striker Chris Wood.
Mr Hartley said the origins of the family remained a mystery, meaning the animals are being treated as normal strays.
He said the club was concerned after they were spotted around the ground during the visit of Senegal - also known as the Lions of Teranga - to the City Ground.
"There were over 25,000 people in the stadium, and apparently mum was just milling about under the stands while people were there, so \[the club\] contacted us," he said.
"They all seemed to be in good health - mum had obviously been feeding them really well, because the kittens were a decent weight, \[and\] mum was herself a little bit skinny, but that happens regularly.
"Clearly the staff had been helping feed them and keep them in good nick."
The animals are all set for an away trip to the vet, where they will be microchipped, neutered and given a thorough medical before being made available for transfer.
Mr Hartley said he hoped the adoption process would be completed before the start of the new season.
"Mum's really friendly - the kittens are still a little bit shy, but we're working with them," he said.
"Obviously we get attached to all mums and kittens, but they're definitely ones we're going to remember."