The South African was unceremoniously dumped last month, just weeks before he was due to lead Australia into an Ashes series for the first time, after a string of poor results and incidents of ill-discipline in the squad.
Arthur is suing for $3.6 million (£2.4m) or reinstatement and more than five hours of talks at the Fair Work Commission in Sydney on Wednesday were aimed at keeping the case out of the courts.
"Although we did not come to an outcome today, I am pretty confident there was enough goodwill in the room for us to continue talks next week," Arthur told reporters after the hearing.
"As I said, all we want is a fair and reasonable outcome and I'm pretty sure we'll achieve that next week."
The affable former Proteas coach said on Sunday he was forced into taking legal action because Cricket Australia officials had failed to get in touch with him after his sacking.
Arthur believes his dismissal was immensely damaging to his reputation and career and has noted that Cricket Australia chief James Sutherland had described him as something of a scapegoat.
Ashes holders England won the second test by 347 runs at Lord's on Sunday to take a 2-0 lead in the five-match series.
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