The matter was raised again as councillors heard new proposals are being examined by the government and one option for muzzling dogs - whose behaviour has previously given cause for concern - in public has been been suggested in an options paper.
And the issue of dogs being kept on leads will be looked at too.
Sinn Fein councillor Oliver McMullan said Moyle Council still allows dogs to roam off leads and unmuzzled whilst on Council property and he said it has "now come to the time to think about that again".
He said the Council has to show leadership and said large dogs can knock small children down and do them harm.
Independent councillor Seamus Blaney wondered how such restrictions would be greeted by an elderly person who just wants to walk a small dog.
Cllr McMullan feared if someone is hurt by a dog on Council property the Council could be at risk of legal action.
SDLP councillor Madeline Black said she does not have a dog but said dog owners want to take them onto the beach and let the dogs off leads.
She said most dog owners she knows are "quite responsible" and she didn't think there was a need to tell them to muzzle dogs to take them along the beach.
Cllr Blaney added most dog owners are "pretty responsible" and the dogs come to their owners when called.
He said other animals running free can cause problems and claimed that as he drove along Ballycastle's Market Street a cat ran out in front him forcing him to slam on the brakes.
Cllr McMullan said he was still not happy that the Council allows large dogs to roam unchecked.
He said he was not talking about the person out walking a small Terrier dog but rather large dogs and he said there is a dog in the area "the size of a calf".
Muzzling dogs would give the public confidence and while he admitted there are responsible dog owners he said the Council has to protect children in their parks and on beaches.
Council Chairperson, Cllr Cara McShane (Sinn Fein), felt muzzling dogs was perhaps too much.
Council Vice-Chairperson, Cllr Robert McIlroy (DUP), said dogs should be on leads to be property controlled in public places.
Independent councillor Randal McDonnell, quipped: "It is the back end (of dogs) you need to muzzle, not the front end."