The incident, during the team's transit from South Africa to Argentina, where the Wallabies are preparing for their final clash of the Rugby Championship, prompted the staff member to return home early, coach Ewen McKenzie said.
"A staff member's flown home but I'm not going to comment on the detail of that," McKenzie said in comments published in the Sydney Morning Herald. "Matters of integrity, you've actually got to protect the people involved until there is some sort of decision. So there's processes going on."
The ARU tweeted: "We are aware of an alleged incident involving Kurtley Beale on a flight from South Africa to Brazil. The matter is being investigated by the ARU's integrity team according to our usual processes.
"We can expect the investigation to be completed by the end of next week."
A team spokesman told the Sydney paper that alcohol had not been directly involved in the altercation, which was restricted to a verbal disagreement between Beale and the staff member.
McKenzie said the issue had distracted from the team's preparations for their clash in Mendoza on Saturday.
"I can assure you I don't enjoy the distraction of these things because they take massive amounts of time," he said.
"I understand better than anyone else because in the end you have to deal with it and make sure it's dealt with."
Beale, who was the Wallabies' starting fly-half for the tournament's opening two matches against New Zealand before being demoted to the bench, has a patchy off-field record and spent time in rehab last year to deal with alcohol issues.
He was slapped with a lengthy ban by his former Super Rugby club Melbourne Rebels last year for coming to blows with the team's captain and another team mate in a boozy incident on a bus while on tour in South Africa.
He was later stood down by the club indefinitely for breaching his drinking restrictions.
Though he returned to the Wallabies fold for last year's British and Irish Lions tour, he embarrassed the team and infuriated senior players by being photographed with sacked former team-mate James O'Connor at a fast food restaurant at 0400 days before the second Test.
Beale's probe has further overshadowed the Wallabies' preparations which have been hampered by their baggage failing to turn up in Argentina and injury clouds over key players.
The Wallabies were over-run by the Springboks in the last 10 minutes last week when the home side scored three tries to record a 28-10 victory in Cape Town.