The court said confidential meetings were "very productive" and that "all agreed we should aim to reach agreement within the time frame of the conciliation process".
The court ordered compulsory arbitration in the case last month after East Timor requested the process over objections from Australia. It wanted plans to finalise the border after 2056 scrapped in response to allegations that Australia spied on East Timor.
At stake is the revenue sharing deal for the 5.1 trillion cubic feet of gas and 226 million barrels Greater Sunrise project, which is operated by Woodside.
The border dispute and low gas prices mean its development is on hold.
East Timor says the boundary should midway between it and Australia, while Australia prefers using its continental shelf.