RNZ correspondent in Fiji, Lice Movono, said the 12 cases were from two households.
"That tells us that people are still moving in and out of each other's homes, people are not maintaining any sort of bubble."
Movements of the 12 people in the past few days included trips to supermarkets more than 30 minutes drive away from their suburb.
"The people from that area were shopping within the city. The supermarkets that were closed today [were] one from their area, but two within the city, the central business district," Movono explained.
"So there is still a lot of unnecessary travel way past neighbourhoods."
She said government ruled out a full lockdown in the next day or two, saying it would give the public adequate advance notice if it decides on one.
Fiji now has 48 active cases, 35 of them locally transmitted, and seven in border quarantine. The source of six cases is under investigation.
According to Movono, panic is growing. At least three supermarkets today were closed today in response to the Covid-19 cases.
"That caused a lot of panic for people who live in and within the city area because these are their main supermarkets," she said.
Meanwhile, Fiji's government has sought to reassure a nervous public on edge in anticipation of a lockdown.
"There's definitely no lockdown in the next day or two. We are going to tell you well in advance; we will give you not a few hours but we will give you days' notice so that not just you but also the government machinery can prepare."