The Senate could refer the eligibility of One Nation senator Malcolm Roberts to the high court as early as Wednesday, after the senator’s repeated refusal to release his personal documents.

But the Greens’ attempt to audit the citizenship status of every MP in parliament will be quashed, because the Coalition and Labor refuse to support it.

A cloud hangs over the eligibility of some MPs in parliament, and the Senate agreed on Tuesday to refer the eligibility of Nationals senator Matt Canavan and Greens senators Scott Ludlam and Larissa Waters to the high court to determine whether they were allowed to stand for election last year.

All three senators discovered in recent weeks that they held dual citizenship when they nominated for the election, putting them in breach of section 44 of the constitution, “any person who is a subject or a citizen or entitled to the rights or privileges of a subject or a citizen of a foreign power” is incapable of being chosen to sit in Parliament.

Ludlam and Waters resigned from parliament on discovering their dual citizenship, but Canavan chose to stay, only standing aside as minister for resources and northern Australia pending further legal advice. Canavan said on Tuesday that he welcomed the Senate’s decision to send his matter to the high court, saying he would not vote in the Senate until the matter was resolved.

The Greens leader, Richard Di Natale, said he would ask the Senate on Wednesday to ask the high court, acting as the court of disputed returns, to consider the eligibility of Roberts, because he has repeatedly refused to release his personal documents clarifying his citizenship status.

New documents cited by Buzzfeed on Tuesday again confirmed that Roberts had been a British citizen in the past, contradicting the senator’s tweet in October 2016 that he had “never held any citizenship other than Australian”.

Any motion to refer Roberts to the high court supported by Labor and the Greens and opposed by the government will require a further four senators to succeed.

Labor and cross-benchers Jacqui Lambie and Derryn Hinch are likely to support the motion, putting it within a couple of votes of success. The Turnbull government has indicated it will not support the motion, citing a lack of evidence that Senator Roberts is in the wrong.

“The onus of proof lies on those who seek to assert they weren’t validly elected to demonstrate that was the case,” Attorney-General George Brandis said. “It’s not a reverse onus of proof on Senator Roberts to demonstrate he was validly elected.”

The motion would still succeed with the support of the Nick Xenophon Team but Senator Xenophon says he is still considering the proposal. If he refused to support it the Greens would need the votes of conservatives Cory Bernardi, Lucy Gichuhi or David Leyonhjelm.

Labor senator Penny Wong said on Tuesday: “I know some in this place have called for the appointment of some body or eminent person to ‘audit’ the eligibility of all senators. “This isn’t an approach the Labor party supports.”