Mr Parker, who is under pressure to explain the NSW CFMEU branch's dealings with alleged crime figure George Alex, is expected to next week face further questions over the leaking of private details of Cbus superannuation fund members to him.

Colourful Sydney business identity Jim Byrnes is expected to appear before the commission on Friday.

Fairfax Media has confirmed that a business owned by Mr Alex, who has been accused in the commission of paying kickbacks to union officials, financially supported a NSW CFMEU fundraiser. The union is a big donor to the ALP.

Evidence uncovered during an investigation commissioned by the tax office reveals that one of Mr Alex's companies arranged for the purchase of tickets on two tables at a union fundraiser.

Details of Mr Parker's recalling come as two more CFMEU officials, Brian "Jock" Miller - a 29-year veteran of the union and senior organiser, Andrew Quirk, have gone public to call for the leaders of Australia's labour movement to act.

Both want senior ALP and Australian Council of Trade Unions leaders to seek an urgent briefing from law enforcement on the alleged "overlap" between certain CFMEU officials and organised crime figures.

Mr Quirk and Mr Miller accuse the CFMEU's national leadership of failing to act on malfeasance.

"There has been a pretty catastrophic failure of governance in the CFMEU from the level of the management committee [in NSW] to the top of the union," Mr Quirk told Fairfax Media.

Mr Quirk and Mr Miller, who are also due to appear on the ABC 7.30 program on Thursday night, urged the ALP and the ACTU to speak out against the victimisation of whistleblowers in the union. Mr Quirk said those who speak out were being "forced out of their jobs and their careers".

"The silence is deafening. If people are really concerned, the way they say that they're concerned within the union movement and within the labour movement about corruption in the labour movement, then why don't the relevant people in the senior ranks of the ACTU and the Labor Party go and seek the relevant briefing from the relevant security authorities and from the relevant police authorities on the state of play?" Mr Quirk said.

"At what stage is somebody going to get up and act like a mature, responsible grown up, and recognise that dealing with criminals … has nothing to do with the labour movement?"

The ACTU is believed to be open to receiving a briefing from law enforcement.

Mr Quirk and Mr Miller join veteran CFMEU officials Brian Fitzpatrick, who left the union last year, and long-time delegate Mario Barrios, in raising serious concerns about the union's activities.

Mr Miller said figures in the CFMEU "seem to be protecting other people just to save their jobs instead of telling the truth".

"We've got a problem and we need to fix it. Either that or the union is going to be decimated," he said.

The pair decided to speak out after the royal commission recently revealed evidence, including phone taps, that appeared to show CFMEU NSW secretary, Mr Parker, supporting a business owned by Mr Alex.

Counsel assisting the commission, Jeremy Stoljar, has also alleged that weekly kickbacks were paid to the union by Mr Alex, who in 2012 and 2013 employed numerous bikies and convicted terrorist Khaled Sharrouf in his construction business empire.

Phone taps played at the commission captured Mr Parker telling another union official he wanted to bash a union whistleblower who was questioning his suspected support of Mr Alex's companies.

Mr Parker also appears to have been caught misleading the royal commission about his knowledge of the leaking of workers' personal information from a superannuation fund to the CFMEU.

The CFMEU's national assistant secretary Dave Noonan has been the face of the union's public response to the allegations, repeatedly claiming that the commission is a politicised witch-hunt that is ignoring corruption or malfeasance by building companies.

Late last year, the CFMEU's national secretary Michael O'Connor commissioned an internal inquiry by former union official turned barrister, Tony Slevin, which found corruption and misconduct allegations inside the union were unfounded.

Several of the report's findings have since been heavily challenged by evidence produced by the royal commission.

Mr Quirk has alleged that Mr O'Connor failed to investigate several allegations he made about the infiltration of criminals into the union in NSW and Victoria.

"I'm saying to Michael, look… you've got a problem in Melbourne and you've got a problem in Sydney, mate, right?  There's no good running away from this."

Last month, Victoria Police assistant commissioner Steve Fontana told the royal commission that some in the CFMEU were involved in criminal activity, but declined to publicly provide evidence to back up his comments, citing the need to protect witnesses.

In a statement released earlier on Thursday, the national office of the CFMEU said the union did not tolerate corruption and urged those with concerns to contact police.

However, the union said the royal commission had yet to provide any "credible evidence" to support allegations of a corrupt relationship between Mr Alex and one of its officials.

The union said its national leadership had taken the allegations about the NSW CFMEU branch seriously and had devoted considerable resources to investigating them.

"What has become clear in recent months is that there is a faction operating within the NSW branch of the CFMEU that is attempting to destabilise the current leadership and using the royal commission as a vehicle to pursue their political objectives."

ACTU secretary Dave Oliver said his organisation had consistently said that criminal behaviour and corruption must be dealt with by the police and the courts.

"There is no place in our movement for criminal behaviour or corruption," Mr Oliver said.