Indonesia's president has suspended co-operation with Australia in combating people-smuggling and other activities, and condemned the country's "Cold War" behaviour following allegations its spies tapped his phone.

Susilo Bambang Yudhoyono announced on Wednesday that co-operation would be temporarily halted in a number of areas, including military exercises and intelligence exchanges.

But the most important area for the new Australian government of Tony Abbott will be people-smuggling, as it tries to halt the flow of asylum-seekers arriving by boat from Indonesia.

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 listen to the phone calls of the president, his wife and ministers in 2009.

Indonesia recalled its ambassador from Canberra earlier this week over the scandal.

Announcing the suspension of cooperation after a crisis meeting at the presidential palace, Yudhoyono said: "For me personally, and for Indonesia, the wiretapping by Australia ... is difficult to comprehend. This is not the Cold War era."

The flap is being seen as the lowest point in a perennially volatile bilateral relationship since 1999, when Australia led a UN military force into the former Indonesian province of East Timor following a bloody independence ballot.

Public display of anger

Yudhoyono said that on Wednesday evening he would send a letter to Abbott to personally ask him for an explanation.

"I am still hoping, and I am sure Australia is still hoping ... that we can still work together after this problem is overcome," he said.

The public display of anger comes before Indonesia's general and presidential elections next year, with the president's ruling party slipping badly in recent opinion polls.

Abbott said he would be respond to the letter "swiftly, fully and courteously" and expressed "deep and sincere regret about the embarrassment" caused by the spying claims. However he once again refused to apologise.

Abbott told parliament on Wednesday that he would do everything he "reasonably can" to repair relations with Indonesia.

He proposed not to "overreact" to Indonesian anger over the eavesdropping allegations, but added that his intention was to "build and strengthen the relationship with Indonesia which is so important to both our countries".

Indonesia already has recalled its ambassador following the reports that Australian spies attempted to listen to the president's cellphone in 2009.