Housing New Zealand also identified vet bills, clearing flea infestations, dog bites and the stress associated with dog complaints from neighbours or the public as other risks.
Under the current policy tenants have to get permission from Housing New Zealand to have a dog on the property and the owner must prove it is registered and vaccinated.
But Hauraki-Waikato MP Nanaia Mahuta said responsible dog owners shouldn't be penalised because of irresponsible dog owners.
"Housing New Zealand needs to take into consideration that responsible dog owners are taking the required steps to ensure they are looking after the dog on their property and that dog isn't a harm to anyone."
She had not heard from tenants in her electorate about the current policy, so could not gauge whether it was working.
Royal New Zealand SPCA president Bob Kerridge said a change in the policy could make it even harder than it already was for Housing New Zealand tenants to have dogs.
"What they are doing is depriving families of the enjoyment of having an animal. I think any measures that deprive people of that enjoyment are not to be applauded in any shape or form.
"Unless they were causing a problem I think that would be very unfair."
He urged Housing New Zealand to talk to the SPCA about its policy review, and suggested it considered the size and breed of dog when writing the rules.
Grandmother vows to keep her P'nut
Becky Kippax's five children have long left home but there's still a pitter patter of tiny feet around her house.
The 60-year-old grandmother lives alone except for her 3-year-old Jack Russell cross P'nut.
The little dog follows her around the house and also acts as a guard dog, alerting the sickness beneficiary to any strangers approaching her property.
"If it wasn't for her [P'nut] my clothes would go off the line," Ms Kippax said.
Ms Kippax said she would be "heartbroken" if Housing New Zealand's policy review meant she would have to get rid of P'nut.
"I would refuse it. They can't take her off me," she said.
She had been living in her Housing New Zealand home on Smith Ave in Huntly for 26 years - raising all her children there. In that time she has also owned three dogs.
Ms Kippax said she was a responsible dog owner, had P'nut registered and kept her within the fenced property.
She had not been informed about Housing New Zealand's dog policy review.