Sydney has witnessed an unabated spike in locally acquired coronavirus cases, and on Tuesday, Australia's biggest city reported five such infections, as authorities urged the public to get tested and increase openness and transparency with contact tracers.

Two of the cases were flagged on Monday, and involved a man who attended a hospital emergency department and his household contact, with investigations continuing into how they contracted the disease, Xinhua news agency reported.

The others were linked to known clusters in the city's west and northern beaches.

Testing rates fell to 14,738 on Monday, and health officials hoped to increase the figure to over 25,000 per day.

"As we see, the disease is still bumbling along in the community and we need to be vigilant about that and that's why I'm pleading for people to come forward and get tested," New South Wales Premier Gladys Berejiklian said.

She added that some people had withheld information from health experts due to a fear of repercussions, assuring the public that the only interest was in stopping community spread and that personal medical information was kept confidential.

"There have been situations where people are afraid to tell us everything because they're worried about getting in trouble or worried about having broken the rules," she said.

"I assure you that that is not the case. Don't be concerned about any judgements, don't be concerned about any retribution. Quite the contrary."