Mr Morrison says there will be a formal investigation into whether Mr Heydon's Companion of the Order of Australia should be revoked if the "incredibly serious" allegations are upheld.

"It's not appropriate to presuppose those processes. That's not the way these things should be handled," he told reporters on Tuesday.

"There should be a proper process to deal with this. There will be.

"They're very serious allegations. They're very concerning. And very disturbing. And on that basis, I would expect those processes to do their job."

Labor is calling for Mr Heydon to be stripped of his honour after an independent inquiry found he sexually harassed six women.

Opposition frontbencher Bill Shorten also wants Mr Heydon to pay back the salary he earned from the royal commission into trade unions.

"Why does he get to keep his AC, and if the matter goes to court or there is further investigation, why does he get to keep all his taxpayer earnings from the royal commission?" Mr Shorten asked.

"This is a time to strip him of all his recognition and get him sorted out."

Mr Shorten was grilled at the 2014 royal commission and Mr Heydon later described him as an evasive witness.

There is no love lost between Mr Heydon and the Labor Party.

He was appointed to the High Court by John Howard and chosen to head the royal commission by Tony Abbott.

Six former associates have come forward with allegations of sexual harassment against Mr Heydon.

He categorically denies the allegations.

"Our client says that if any conduct of his has caused offence, that result was inadvertent and unintended and he apologises for any offence caused," Mr Heydon said through his lawyers.

Three of the women are pursuing claims for compensation.

Their lawyer, Josh Bornstein from Maurice Blackburn, said the firm would first try to negotiate an outcome with Mr Heydon's lawyers.

Failing that, they will pursue the matter with the Australian Human Rights Commission, and then through the Federal Court if allowed.

Labor leader Anthony Albanese said his thoughts were with the brave women who came forward with complaints, including those who had left the law.

"I would find it remarkable if the appropriate authorities aren't looking very closely at this and if there's not further action," he told reporters.

Chief Justice Susan Kiefel said she was ashamed the alleged harrassment could have happened at the High Court.

"The findings are of extreme concern to me, my fellow justices, our chief executive and the staff of the court."

The independent inquiry report prepared by former inspector-general of intelligence and security Vivienne Thom has been provided to the complainants and Mr Heydon.

The inquiry report makes six recommendations on which the High Court has acted.

These include establishing a human resources policy relevant to the personal staff of justices and better induction processes for associates.

Australian Associated Press