President Susilo Bambang Yudhoyono, the first lady and Vice-President Boediono were reportedly amongst those targeted.
The documents leaked by Edward Snowden were published by broadcaster Australian Broadcasting Corporation (ABC) and the Guardian newspaper.
Indonesia said Australia should “urgently” clarify the spying claims.
The report is the latest in a series of spying allegations that have strained relations between the two allies.
On November 1st, Indonesia summoned Australia’s ambassador amid reports that Australia’s Jakarta embassy was used as part of a US-led spying network in Asia.
The latest leaked document showed that Australia spy agencies named Susilo Bambang Yudhoyono, the first lady, Vice-President Boediono and other senior ministers as targets for monitoring, the reports said.
The presentation from Australian spy agency the Defense Signals Directorate (now known as the Australian Signals Directorate) showed that agencies attempted to listen to Susilo Bambang Yudhoyono’s calls at least once, and tracked calls made to and from his mobile phone, in August 2009, the ABC and the Guardian added.
The news organizations published slides from the presentation, which appeared to show a list of Indonesian “leadership targets” and the handset models used by each target, as well as a diagram of “voice events” of the Indonesian president in August 2009.
One slide entitled “Indonesian President voice intercept (August ’09)” appeared to show an attempt to listen to the content of a phone call to Susilo Bambang Yudhoyono.
On Monday, responding to questions in parliament, Australian PM Tony Abbott said: “The Australian government never comments on specific intelligence matters.”
He added: “I will never say or do anything that might damage the strong relationship and the close co-operation that we have with Indonesia, which is all in all, our most important relationship.”
Last week, commenting on the earlier claims, PM Tony Abbott had described the term spying as “kind of loaded language” and suggested that “researching” would be more appropriate.
On Monday Indonesian presidential spokesman Teuku Faizasyah, said: “[The] Australian government urgently needs to clarify on this news, to avoid further damage.”
“The damage has been done,” he added.
Indonesia has publicly voiced anger over previous allegations of Australian spying.
Vice-President Boediono, who like many Indonesians goes by only one name, said last week that the Indonesian public was “concerned” about the spying allegations.
“I think we must look forward to come to some arrangement which guarantees that intelligence information from each side is not used against the other,” Boediono said.
Australia and Indonesia are key allies and trading partners.
Australia requires Indonesia’s co-operation on the asylum issue, as many asylum seekers travel via Indonesia to Australia by boat, but there are tensions on the issue.
Earlier this month, Indonesia declined an Australian request to receive a boat of asylum seekers whose vessel, bound for Australia’s Christmas Island, had got into trouble after it departed from Indonesia.
The reports are amongst the series of documents leaked by Edward Snowden, who has been granted temporary asylum in Russia and is wanted in the US in connection with the unauthorized disclosures.
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