The Senate has voted to investigate the circumstances surrounding Peter Dutton’s decision to use ministerial intervention powers to save two foreign au pairs from deportation.

The visa status of the two unknown young women has been in the public spotlight since March, when it was revealed the former home affairs minister granted them visas on public interest grounds in 2015.

Dutton stood aside as minister on Tuesday after challenging prime minister Malcolm Turnbull for the leadership of the Liberal party.His backers circulated a petition late on Wednesday in an attempt to trigger a second leadership spill.

Labor senator Louise Pratt, who chairs the Senate legal and constitutional affairs references committee, moved a motion for a snap inquiry into “allegations concerning the inappropriate exercise of ministerial powers with respect to the visa status of au pairs”.

At noon on Thursday the Senate voted 34 votes to 28 to set up the inquiry, which is due to report by 11 September.

A parliamentary inquiry would give legal protection to witnesses, could demand paper trails and compel people to attend, including ministerial staffers and public servants.

However, there are no guarantees a Senate inquiry will be able to get to the bottom of the matter.

In 2003, a Senate inquiry examined the so-called cash-for-visa allegations that then-immigration minister Philip Ruddock intervened in visa cases in exchange for donations to the Liberal party of up to $100,000 from the individuals or their associates.

That inquiry’s final report released in March 2004 stated the committee had been denied critical case files and “was left in no doubt that it was obstructed” in carrying out its full task.

Earlier this month, Guardian Australia revealed the Department of Home Affairs had spent $10,000 in taxpayer cash fighting a legal battle to keep documents secret about the au pair visa decisions.

In the first case, an au pair whose visa was cancelled at Brisbane’s international airport in June 2015 was able to make a phone call and within a couple of hours Dutton approved a new visa.

According to two sources this au pair is a French citizen. In November the same year, Dutton defied written concerns from his own department that granting a visa to a second au pair was of “high risk” because she had previously been warned about work restrictions.

Department sources told Guardian Australia the second au pair is also from a western European country.

Dutton has previously insisted he doesn’t know the two au pairs and they didn’t work for his family.

“For the wider record, I do not personally know the individuals concerned, nor does my wife. They have never been associated with us in any way. We have never employed an au pair,” he said in March.

“There were two young tourists who had come in on a tourist visa and declared … [they] intended to perform babysitting duties while here. The decision that was taken … that those two young tourists would be detained and that they would be deported. I looked into the circumstances of those two cases and I thought that inappropriate.”

Dutton has previously refused to answer specific questions about the identity of the au pairs’ employers.