Mr Turnbull has issued the invitation for 2018 during a special address at the ASEAN summit in Laos.
Among the summit's key developments was a brief encounter between United States president Barack Obama and the Philippine president who had called him a "son of a whore."
The South China Sea dispute and concerns over regional security have continued to dominate the ASEAN summit in Laos.
The Philippines, which won a case against China in the International Court of Arbitration in July, has released new images from the South China Sea.
It says the pictures show China preparing to possibly reclaim more land as well as building infrastructure on another disputed territory.
China denies it has any such intention.
Prime Minister Malcolm Turnbull has told ASEAN leaders all parties in the South China Sea must exercise restraint.
"Differences should be settled peacefully in accordance with international law. And until they are, we urge all sides to act with restraint and avoid any steps which will create, or add to, tensions."
Malcolm Turnbull made those comments as he sat down for the first ASEAN-Australia dialogue, which gives Australia a direct audience with the 10 nations in the region.
It has been on Australia's diplomatic wish-list for decades.
The Prime Minister has invited the leaders to a special summit in Australia in 2018.
He says it would draw regional partners closer together to tackle issues such as the South China Sea.
"A special summit in Australia would deepen our economic partnership through closer links between ASEAN and Australian businesses in the private sector and bolster our strategic partnership."
Another issue at the forefront of the summit and sideline meetings is regional security and terrorism.
Malcolm Turnbull says, as the self-proclaimed Islamic State, also known as ISIL, continues to lose ground in Syria and Iraq, it will focus more closely on South-East Asia.
He says he wants much greater cooperation on intelligence-sharing to keep everyone better informed on the recruitment and movement of terrorists.
"We know that ISIL wants to expand its presence in South-East Asia, and this is of concern to us all. The latest ISIL propaganda is further evidence that, as it faces increased pressure in Syria and Iraq, it will attempt to seed fear and division in our communities. Our updated joint declaration is a significant signal of collective political will to share our expertise and stay ahead of this evolving threat."
ASEAN leaders joined other partners for individual meetings like the one hosted by Mr Turnbull before moving into the East Asia Summit.
That includes ASEAN's North Asian neighbours China, South Korea and Japan and dialogue partners Australia, the United States and New Zealand.
The leaders have also attended a gala dinner to mark the start of the summit.
At the dinner, the Philippines confirmed its president, Rodrigo Duterte, had finally met US president Barack Obama.
The men were initially scheduled to meet on Tuesday, but Mr Obama cancelled after Mr Duterte called him a "son of a whore."
Mr Duterte had also threatened to swear at him if he brought up certain domestic issues.
But Philippine foreign secretary Perfecto Yasay says the pair finally crossed paths for a brief chat, although he would not disclose the details of the conversation.
"Yes, I will confirm that they met. They met in the holding room, and they were the last persons to leave the holding room. I would not be able to say at this time how long they met and talked, but I am informed very reliably that the meeting was very cordial, and I guess the implications of this would be that it all streams from the fact that the relationship between the Philippines and the United States is firm."
Another leader on the attack was New Zealand prime minister John Key, who has hit out at North Korean leader Kim Jong-un.
After meeting with Malcolm Turnbull for talks, Mr Key said the threat from North Korea was one of the biggest issues on the agenda.
"All of us are concerned by the actions that we see from a leader who is highly unpredictable and is building greater capability and undertaking more tests, and, from what we can see, those tests are becoming more successful. The range is further, and that's challenging for regional security in a way that we haven't really seen in other areas. We sit back with a guy that we just don't know what he's going to do next. His behaviour and the way he's treated people, if you believe the media reporting of that, not only is it abhorrent, but it's also ... well, it's barbaric."
Also in attendance at ASEAN was United Nations secretary-general Ban Ki-Moon, who urged leaders to ratify the Paris agreement on climate change.
Last December, nearly 200 countries agreed on a binding global compact to keep global temperature increases to what it called "well below" two degrees Celsius.
Ban Ki-Moon says, although 180 countries have since signed the agreement, 55 nations need to formally ratify the treaty to put it into legal effect.
"Twenty-seven countries, as of now, have ratified the agreement, accounting for some 39 -- a little over 39 -- per cent of greenhouse gas emissions. We need that 28 more countries ratify, and accounting for a further 16 per cent of greenhouse gas emission, to bring this agreement into force."
