Calls by Senate crossbenchers for the government to broaden its focus on corruption beyond the construction sector trade unions have been supported by a new poll which finds almost two-thirds of voters support a national Independent Commission Against Corruption, or ICAC.

A national Research Now poll, commissioned by The Australia Institute, found only 11 per cent support for an anti-corruption body for the building industry, while 65 per cent supported a national ICAC.

Prime Minister Malcolm Turnbull has recalled Parliament for April 18 and given the Senate crossbench three weeks to pass two bills cracking down on trade union misbehaviour or face a double-dissolution election on July 2.

Senate crossbenchers including Senator Ricky Muir, Dio Wang, Jacqui Lambie and Glenn Lazarus. Alex Ellinghausen

The poll of 1412 voters found 33 per cent agreed with this strategy, 22 per cent disagreed and a significant 36 per cent were unsure.

With Labor and the Greens opposed to the bills – which would re-establish the Australian Building and Construction Commission, and establish a Registered Organisations Commission to oversee unions and employer groups – Mr Turnbull's threat is aimed at the eight-strong independent Senate crossbench.