Foreign Minister Julie Bishop told reporters in Cairns: "The current laws safeguard the privacy of Australians but also provide us with an opportunity to keep Australians safe."
The story ran in Sunday editions of News Corp newspapers and cited correspondence between the head of the Department of Home Affairs, Mike Pezzullo, and Defence Secretary Greg Moriarty.
It said the ASD was seeking powers to monitor Australian citizens for the first time and such powers would allow the agency - which has so far had an external role - to secretly access emails, bank records and text messages.
A denial also came in a joint statement from Pezzullo, Moriarty and ASD director Mike Burgess.
"There is no proposal to increase the ASD's powers to collect intelligence on Australians or to covertly access their private data," they said.
"We would never provide advice to government suggesting that ASD be allowed to have unchecked data collection on Australians – this can only ever occur within the law, and under very limited and controlled circumstances."
Labor shadow attorney-general Mark Dreyfus called for an investigation into how such correspondence had leaked.
"The documents described in the media appear to be extremely sensitive and divulge information about one of Australia's key security agencies," he wrote in a letter to Prime Minister Malcolm Turnbull.
At the moment, only the Australian Federal Police and ASIO can access the information of Australian citizens and only after obtaining a warrant from the attorney-general.