The Large Breed Dog Rescue in Greenhithe, Kent, said it turned down 25 American XL Bullies in the past week and urged people not to panic.
Jo Woods, a volunteer at the rescue centre, said she had been offered about 25 bullies a week, with owners telling her they would put their pets down if there was no space at her shelter.
“Over the last week, I have turned down 25 dogs and that’s just me. My coordinator also gets contacted by specific rescues as well. It’s just nuts,” she told Kent Online.
The ban follows a series of attacks this year, including two American XL Bully dogs alleged to have killed father-of-two Ian Price in Staffordshire.
Ms Woods said some people had bought the “wrong” type of dog during the Covid-19 pandemic, which has now led to violent teenage and untrained dogs being left in homes.
Campaign group Save Our Seized Dogs said it had also been “inundated” messages from XL Bully owners asking for help, adding that the ban would do “more harm than good”.
Campaigner Jayne Dendle said that irresponsible dog owners “don’t care” what happened to their dogs and they would just dump them after the ban, making the situation worse.
In 2023, over a third of recorded dog attacks were by American Bully breeds including the Bully XL, according to pressure group Bully Watch. The group said 341 out of 841 dog attacks it recorded in 2023 were by American Bully breeds.
Around half of all American bullies bred in the UK are linked to one dog known as “Killer Kimbo”, the pressure group found.
It also discovered that dogs related to the animal, raised in Los Angeles, United States, were responsible for at least 10 violent incidents around the world.
Announcing the ban, Mr Sunak said: “The American XL bully dog is a danger to our communities, particularly our children.
“I share the nation’s horror at the recent videos we’ve all seen. Yesterday we saw another suspected XL bully dog attack, which has tragically led to a fatality.
“It is not currently a breed defined in law, so this vital first step must happen fast. It is clear this is not about a handful of badly trained dogs, it’s a pattern of behaviour and it cannot go on.”