After years of political brawling, two private member's bills will be introduced.

While a vote is months away, Labor backbencher Stephen Jones will present his bill to legalise gay marriage, while Greens MP Adam Bandt and Tasmanian independent Andrew Wilkie will jointly sponsor a second.

However, Mr Wilkie will move that religious ministers not be obliged to perform same-sex ceremonies.

Several other bills on gay marriage will be introduced this year, with Greens senator Sarah Hanson-Young seeking to present a bill in the upper house.

Meanwhile, the federal opposition will continue its attack on the Gillard government in budget estimates hearings this week.

Parliamentarians enter the second week of the sitting period with opposition members planning to continue their attack on the government's weaknesses.

The Australia Day security debacle and the Craig Thomson investigation will both come under scrutiny.

Tony Hodges, the Prime Minister's former staff member at the centre of the fracas, will not appear as staff are not compelled to appear before Senate committees.

The government will seek to demonstrate its diligence.

Prime Minister Julia Gillard will attempt to push through changes to means test the private health insurance rebate after securing the vote of the crossbenches.

The plan is forecast to save $100billion over the next 40 years and Labor sees it as a massive boost to its economic credentials as it attempts to bring the budget back into surplus by May. The plan also involves an increase in the Medicare levy surcharge for Australians without private health cover.

Other legislation for debate this week includes appropriation bills signalled in the mid-year economic fiscal outlook in November, paid parental leave and flexible work hours.