Crossbench senators key to the success of the Turnbull government's industrial relations reforms are increasing the pressure to release a secret report produced by the trade union royal commission.

Senators Glenn Lazarus, Jacqui Lambie and Ricky Muir last year voted against bills to reinstate a Howard-era building watchdog and to impose stricter penalties on union officials who breach their duties, with Labor and the Greens. The Senate twice rejected the latter bill, leaving open a potential trigger for a double-dissolution election.

The Coalition will try again this year to pass both bills, following the Royal Commission into Trade Union Governance and Corruption hearings. The commission released a five-volume final report last month.

Commissioner Dyson Heydon found that union misconduct was "widespread" and "deep-seated".