Prime Minister Scott Morrison faces his second week of parliament without an absolute majority in the lower house, and a potential headache over Peter Dutton’s eligibility.

The Greens and Labor want to refer the Home Affairs Minister to the High Court to test whether he is breaching the constitution over his family financial interest in two childcare centres.

‘I have taken advice in relation to my position, which put the question beyond doubt,’ Mr Dutton told parliament on Thursday.

Section 44 of the constitution disqualifies anyone who has a ‘direct or indirect pecuniary interest’ in any agreement with the Commonwealth.

The opposition parties have been talking with Liberal MPs about voting with them to refer Mr Dutton to the High Court, meaning Mr Morrison has to decide if he will do it himself or lose a vote.

Greens leader Richard Di Natale says some Liberal MPs appear to be ready to vote for a High Court referral, perhaps in revenge against Mr Dutton.

‘I don’t mind how they get there, as long as they land in the right place,’ Senator Di Natale told reporters.

But the prime minister can rest easy knowing he won’t lose a vote of no confidence in himself, with crossbench MPs promising to back him at least until Malcolm Turnbull’s replacement is chosen.

Mr Dutton also faces another headache, with a Senate report into his use of ministerial powers to help two au pairs due on Wednesday.

He could face the High Court referral or a potential no-confidence motion after that.

The Senate will debate laws cracking down on tax avoidance, and a bill to ensure the ABC covers rural and regional audiences, and is ‘fair and balanced’.

An inquiry report into the Trans-Pacific Partnership, which Labor has already promised to support through parliament, is due on Tuesday.

Debate on the TPP will continue on Monday, with Labor MPs under pressure from unions to vote against it.

But Labor’s trade spokesman Jason Clare says the party will support the deal, and look to amend it under a future Labor government.

Lower house MPs will also continue debating modern slavery laws, changes to the My Health Record system, and reforms in the Family Court.