Bridges said it should be led by someone such as a High Court judge, agreed by the National Party, and with the best forensic expertise.
"The reason I think this is very important isn't about me, I don't think it's about the opposition. I think it's ultimately about the fact that a leak such as this potentially affects all Members of Parliament."
Bridges was confident the leaker was not one of his MPs, despite confirming earlier that all would have received the information.
"I very much doubt it. I'm incredibly confident it's not going to be a National MP."
But if it was, there would be consequences, he said.
"A leak from Parliament, involving Members of Parliament, has to be taken very seriously."
Mallard told reporters today that the document used by Newshub was not in the same format as that given to the parliamentary library or to his office ahead of public release.
"I have indicated that to a number of parties and am going to take 24 hours and take soundings about whether there needs to be a further, forensic examination as to the source of the document which was used last night," he said.
"It doesn't reflect well on the organisation as a whole but I prefer to wait for 24 hours now before making any further comment."
Deputy Prime Minister Winston Peters was clear where he thought the leak had come from, saying that of the 56 members of the National Party caucus, 55 of them were suspect.
"They're the ones who would know, not us. We never see that information, nor any other party. No party is getting any other party's information on that matter," he told reporters.
"It just shows you that the happy family that they portray themselves [as] is not quite true."
Bridges has stopped short of blaming the Government directly but said they were looking for distractions from their own problems.
"I think you've got to say the Government is looking for every distraction they can rather than focusing on the things they should be – plummeting business confidence, economy in downturn, strikes."
Prime Minister Jacinda Ardern has denied any of her MPs were involved.
"We've sought assurances from Ministerial Services who manage this information, that none of the opposition's numbers were shared with anyone but them and they've given us that assurance."
Ardern said she could categorically rule out anyone in Labour having access to the information, including the whips.
"The only groups, as I understand, who will have had access are the opposition themselves and the Speaker," she told reporters.
Bridges has defended his expenses, which come after his nationwide tour to raise his profile, saying he had been working hard to get to the regions and understand their issues.
Ardern said earlier that when she saw her expenses after first becoming party leader, she was surprised how high the figure for surface travel was.
"I remember wanting to change my behaviour as a result of that. These numbers, sometimes, when you think you're just out doing your job you suddenly receive an expense and some of the figures can be extraordinary. It certainly threw me the first time that happened," she told Newstalk ZB.
Ardern's surface travel for the first quarter, which included her time as opposition leader, totalled $82,795.
