The Indonesian president Wednesday suspended cooperation with Australia over the sensitive issue of people smuggling, denouncing Canberra's "Cold War" behaviour following allegations its spies tapped his phone.

Susilo Bambang Yudhoyono announced that cooperation with its southern neighbour in a number of areas would be temporarily halted, including military exercises and intelligence exchanges.

But the most important for the Australians will be in the key area of people smuggling, as Canberra seeks to stem a flow of thousands of asylum seekers arriving by boat from Indonesia.

Yudhoyono said that "coordinated military cooperation" between Jakarta and Canberra, which includes joint work on people smuggling, would be halted "until everything is clear".

"For me personally, and for Indonesia, the wiretapping by Australia... is difficult to comprehend," an angry Yudhoyono told reporters.

"This is not the Cold War era."

Cooperation with Australia on exchanging intelligence and sharing information would also be suspended as well as joint military exercises, the president said.

He was speaking after a meeting at the presidential palace in Jakarta with Indonesia's ambassador to Australia who was recalled earlier this week over the scandal.

It was the latest angry outburst from Indonesia over reports, based on documents leaked by US intelligence fugitive Edward Snowden, that Australian spies tried to tap the phones of the president, his wife and ministers.

On Tuesday, Yudhoyono publicly lambasted Australian Prime Minister Tony Abbott on Twitter for what he said was a lack of remorse over the allegations, first reported in Australian media.

Foreign Minister Marty Natalegawa earlier said Indonesia would downgrade ties with Australia, telling reporters: "Like taps, we are closing off areas of cooperation one by one."

Asylum seekers are a flashpoint issue in Australia, and stopping the influx of thousands of would-be refugees who board boats in Indonesia is a priority of Abbott's new government.

He partly won power in September with a series of hardline policies to stem the flow of asylum boats, including turning vessels back to Indonesia when it is safe to do so.

Yudhoyono's decision to suspend cooperation in several areas came after Abbott again refused to apologise Wednesday over the scandal.

"I do understand how personally hurtful these allegations have been, these reports have been, for him and his family," he told parliament.

"I do note there have been allegations and even admissions in the past on this subject, people didn't overreact then and I certainly don't propose to overreact now."

Abbott appeared to be referring to an admission by Jakarta's former intelligence chief to similar spying operations by Indonesia in the past, reported in the Australian media Wednesday.

The Australian and the tabloid Sydney Daily Telegraph both cited comments from a 2004 television interview with Indonesia's retiring intelligence chief Abdullah Mahmud Hendropriyono, in which he said all governments tapped each other's communications.

He admitted that Jakarta had eavesdropped on the phone calls of Australian politicians, had tapped Australian civil and military communications and even bugged the Australian embassy in Jakarta during the East Timor crisis in 1999.

At the time, no apology was sought by then Australian prime minister John Howard.

The leaked documents, reported by the ABC and the Guardian newspaper, showed that Australia's electronic intelligence agency tracked Yudhoyono's activity on his mobile phone for 15 days in August 2009, when Labor's Kevin Rudd was prime minister.

At least one phone call was reportedly intercepted.

The list of tracking targets also included Yudhoyono's wife Ani, Vice President Boediono -- who was in Australia last week -- former vice president Jusuf Kalla, the foreign affairs spokesman, the security minister and the information minister, the reports said.