The kittens, thought to be two different litters, were left in a cat box outside Clive Church, near Wem.
Local people telephoned the nearby Grinshill Animal Rescue and they were collected by the centre.
Jenny Martinez from the rescue centre said: "The catch on the carrier was broken, it was lucky that none of them had escaped as they would never had survived."
"Five of the kittens were five to six weeks old the others four to five weeks old."
"A couple of the kittens were quite poorly with temperatures and had to go on antibiotics and the younger ones are having to be bottle fed."
A week after they were abandoned Jenny said that they were all doing well.
"We are still bottle feeding the tiny ones but they are all putting on weight. We weigh them every day to ensure they are ok. They are all using the litter trays and they all have different personalities. It's hard work but it is good to see them growing."
Jenny, who runs the centre with daughter, Kate, said she wished that the mothers had been left as well.
"We would simply ask the owners to have them neutered, she said.
"Once these kittens are about seven weeks old the mothers can get pregnant again and there will be more unwanted litters.
"There is no excuse for not neutering cats. People who can't afford it can get help from the cat protection league and other charities with vouchers to take to the vet.
"We will help if people really can't afford it."
The centre has been inundated with offers to home the kittens and Jenny said good homes will be found for them among about 60 enquiries that the centre has had.
"We will keep then here until they are about nine weeks old and had their first vaccinations, been neutered, microchipped and wormed.
The centre says that although the kittens are getting through a lot of milk powder it is the vet bills that cost a fortune," she said.
"We would be so grateful for donations towards the vet costs."
People can find out how to help the centre by going to its website grinshillanimalrescue.co.uk.
While the rescue centre has a small cattery, it specialises in caring for and rehoming unwanted dogs.
"At the moment we only have larger dogs and those who will probably have to stay with us because of their behaviour," Jenny said.
She said that the pandemic had had a devastating effect on the centre's fundraising.
"We lost about £28,000 in fundraising that we would have expected to have brought in, over the lockdowns. We would normally raise money by having stalls at events and with the dog shows that we run. But nothing has been happened since 2019.
"We still have all out outgoings event though we have had not income at all."
The centre will hold its first fundraiser on July 4 when there will be an afternoon tea event based at Clive Village Hall.
"People must order their afternoon tea for £8 and they can then go and have a picturesque walk on Grinshill and enjoy a picnic. We will be supplying take away teas and coffees as well. Of course then can take the afternoon tea home if they prefer."
"We are having to think of thinks we can do within restrictions."