A row over the role a senior ministerial staffer played in taking down a food ratings website has seen Prime Minister Tony Abbott become involved.

Mr Abbott fended off questions from Labor frontbencher Tony Burke in parliament on Thursday, opting to take some on notice while he sought more information.

Mr Burke was keen to link the prime minister to the appointment of Alastair Furnival, chief of staff to assistant Health Minister Fiona Nash.

Senator Nash has been under pressure to explain what role Mr Furnival played in the website being taken down just 20 hours after it began operating.

Mr Furnival retains a shareholding in the lobbying firm Australian Public Affairs, operated by his wife Tracey Cain, which represents food industry opponents of the website including Kraft and Cadbury.

Labor insists there has been a serious breach of the ministerial staff code of conduct.

Mr Burke quizzed the prime minister about whether Senator Nash had revealed Mr Furnival’s apparent conflict of interest to a meeting of the Australia and New Zealand Regulation Ministerial Council she chaired in December.

He also wanted to know why, given the “serious conflict of interest”, the government had not reinstated the website.

Mr Abbott said the website had been taken down because it was not ready to go.

“That’s why it wasn’t proceeded with,” he said.

The prime minister took Mr Burke’s other questions on notice, telling parliament: “If there is anything to come back to him on, I will.”

In the upper house, Senator Nash was asked nine questions about Mr Furnival.

The Senate also was told a senior health department official with responsibility for the website, Kathy Dennis, was demoted six days after it was taken down.

The minister said the first she knew about the demotion was in an internal email and directed questions about it to the health department secretary.

She declined to reveal the date the prime minister’s office approved the appointment of Mr Furnival and whether the shareholding was disclosed at the time and was on his private interest statement.

Senator Nash accused the opposition of “imputing impropriety where none exists”.

“All the information about my chief of staff was given to the prime minister’s office in accordance with appropriate timing,” she said.

Later, Labor’s health spokeswoman Catherine King told parliament the minister’s explanations were not true.

When the website went live last week, all stakeholders were emailed to tell them it was online, she said.

“The website … was not a draft,” she told the lower house, adding that she, public health officials and several state health ministers had seen the site.

“It was finalised and agreed by all of the people, all of the states and territories developing the system that it was ready to go.”

Ms King also said an explanation that a ministerial council wanted a further cost-benefit analysis done was also untrue.

A communique from the council’s December meeting showed it was Senator Nash alone who had directed her department to do the extra work.