The calls appear to originate from local numbers.
Once answered, a recorded message tells recipients that they've won a free trip from Shoppers Drug Mart.
Individuals are then connected to a live operator who asks for their credit card information.
Theresa Larabee of Moncton said the conversation immediately raised red flags for her.
“He said, ‘Okay, well I can't talk anymore, we're going to lunch,’ and he hung up. And then I tried to call back and there was no answer," she said.
Trisha Scribner of Moncton received the calls several times.
She said the federal government should be doing more to protect consumers.
“In the states they have all this stuff, they have locks that you can put for free on your phone, if someone's calling your private [number] they'll have to explain who they are. Numerous things like when you say ‘Don't call me back’ — if they call you back there's a lawsuit in the states. In Canada they just don't care,” said Scribner.
Lana Gogas, a spokesperson for Shoppers Drug Mart, said the company would never ask for credit card information over the phone.
“But we are just telling people that this is indeed fraud. Shoppers Drug Mart would never ask for credit card information over the phone, ever. The contest that they are referring to does not even exist at Shoppers Drug Mart currently,” she said.
“If anybody is worried about or has provided personal information they [should] just contact their credit card provider immediately.”
RCMP in New Brunswick are referring scam victims to the Canadian Anti-Fraud Centre.