The prime minister, Malcolm Turnbull, has denied the Senate is “in chaos” despite refusing to commit to bring key industrial relations bills to a vote this year following questions over two crossbenchers’ eligibility.

Mitch Fifield, the manager for government business in the Senate, told ABC AM the government was “absolutely committed” to the Australian Building and Construction Commission bill but also refused to commit to bring it to a vote this year.

The government’s legislative agenda has been thrown into doubt by the resignation of Bob Day, the referral of his eligibility to the high court, and a similar referral of One Nation senator Rodney Culleton to the court over a conviction for larceny at the time of his election.

The draft program for the Senate, sitting next week, shows that plebiscite-enabling legislation is the first piece of government business, followed by appropriations bills, counter-terrorism legislation and bills dealing with the backpacker tax and passenger movement charge.

The ABCC and registered organisations bills, which were the double-dissolution triggers for the 2 July election, do not appear.

At a media conference in Sydney on Thursday, Turnbull said “there is no chaos” in the Senate.

“Seriously, there are two crossbenchers whose eligibility will be considered by the high court. The rest of the crossbench are unaffected,” he said.

Turnbull said it was desirable that both matters be dealt with quickly although it was a matter for the court.

“The constitution will be upheld. We are getting on with the business of government.”

Turnbull said the government would bring the ABCC and registered organisations bills “when we believe there is a majority that will support it and on terms that we will accept”.

Asked if the government would bring the bills to a vote this year, he replied: “It is important that we commit to a vote that we can win in the Senate.”

The prime minister explained the legislation in the Senate next was “very time-sensitive legislation, most notably the same-sex plebiscite bill which we need to get passed if we’re to have a vote in February”.

Earlier, Fifield told ABC AM the government would “deal with the numbers in the Senate as they are”.

Culleton has suggested he will abstain on major bills while his eligibility is under challenge, although he is still entitled to vote.

On Wednesday the attorney general, George Brandis, said Culleton had confirmed in writing to him and the president of the Senate that he “doesn’t propose to exercise his vote on any contentious legislation”.

Day’s vacancy cannot be filled until the challenge of his eligibility is complete, because it will determine whether Family First will automatically fill the vacancy or a recount is required.

The Senate shakeup leaves the government needing eight out of nine crossbenchers to supports its legislation when Labor and the Greens oppose it, a more difficult task than the nine of 11 crossbenchers it had needed before Day and Culleton’s eligibility was called into question.

Fifield said the Senate is a “many-splendoured thing” and the government had a full legislative agenda next week, including the plebiscite bill and appropriations bills.

Asked if the government would introduce its two industrial relations bills, Fifield said they were “both before the parliament”.

Fifield said the government considers its legislative agenda “one week at a time” and refused to commit to bring them to a vote before the end of the year. There are just three sitting weeks left this year, beginning on Monday, then 21 November and 28 November.

“[The] registered organisations and ABCC bills remain incredibly important parts of our legislative program and we will be seeking to legislate those,” he said.

Brandis told ABC’s 7.30 program on Wednesday the government would be “undeterred in prosecuting its agenda” in the final weeks of parliament.

Culleton was a critic of the ABCC bill, although others in One Nation are inclined to support it and a party spokesman has insisted they will vote as a bloc.

Culleton’s first conviction was annulled after the election, but his eligibility is still under question because he may not have been eligible at the time he nominated and was elected. He has since pleaded guilty to the larceny charge, but no conviction was recorded.

On Wednesday Brandis claimed he had no knowledge that finance department officials had warned the Abbott government against allowing Day to move his electorate office to a building he had owned.

Asked about warnings from the finance department in 2014, Fifield said the issue at the time was the location of his office and constitutional issues subsequently discovered were a “separate matter”.

Fifield said that as soon as the special minister of state, Scott Ryan, became aware of issues with Day’s eligibility he sought legal advice and terminated the lease on Day’s office.