Australians should be given the chance to repeal the law that bars dual citizens from serving as MPs and Senators, a parliamentary committee has recommended.

As another four MPs face byelections sparked by a bombshell High Court ruling on citizenship last week, the report warns the problems caused by Section 44 “are not going away”.

“These issues have to be fixed some time. The committee considers that time is now,” the report said.

Commissioned by Mr Turnbull after the High Court disqualified five parliamentarians, including former deputy prime minister Barnaby Joyce, the report calls for a referendum to repeal section 44 of the constitution.

Another option was to consider amending the constitution to give Parliament the power to determine who should be disqualified from public office, the Joint Standing Committee on Electoral Matters’ report says.

Special Minister of State Mathias Cormann responded immediately on Thursday, saying the government would consider the recommendations but was “not inclined to pursue a referendum … at this time”.

“As the committee itself recognised, successful constitutional change is challenging and making the case would take more time than is available prior to the next general election,” Senator Cormann said.

While also conceding there is not enough time to resolve the citizenship problem before the next election, the report suggests the government move to improve compliance with existing laws.

That includes supporting the Australian Electoral Commission to provide more education and support, particularly to independents and minor parties, and discussing a new process of expedited or automatic citizenship renunciation with foreign governments.

Senator Cormann said the government was already taking active steps to “minimise the risk of a recurrence of the eligibility issues that have arisen in the 45th Parliament”.

“We intend to move now to improve the existing candidate nomination process for elections in accordance with the relevant unanimous [committee] recommendations on these matters,” he said.

All candidates would need to provide information to prove they were not dual citizens, in a process similar to the citizenship register set up in Parliament last year.

The committee’s report says that if Section 44 was repealed in a referendum, the government would consult with the public to determine contemporary expectations for politicians’ eligibility that would then be legislated.

“If it is decided that dual citizenship is not a barrier, then future legislation can set out sensible alternative ways of ensuring allegiance,” the report says.

Senator Reynolds said on Thursday: “The committee makes no judgement on the dual citizenship issue itself.”

“The question of whether or not the application of these rules meets contemporary Australian expectations is a different matter altogether and is one for Australians to ultimately determine.”

Though the committee’s inquiry was prompted by the citizenship crisis that has prompted 15 MPs and Senators to either resign or be kicked out of Parliament by the High Court, the report calls for more sweeping changes to the constitution.

It labels as unfair the law that prohibits people in public sector employment – such as teachers and nurses – from nominating for Parliament.

It also argues that politicians should only be forced to divest from financial conflicts of interests when they are sworn in, rather than when they nominate.

Opposition Leader Bill Shorten has said he would prioritise referendums on indigenous recognition and the republic before a vote on Section 44.

The Courier Mail reported on Thursday that two of the three Liberal candidates in the running to contest the byelection in Longman, north of Brisbane, faced dual citizenship questions.