"We will... create a new body, a different body that will be separate," Sir Stuart - a member of the Commons Estimates Committee, told the BBC.
"I would hope that once this body is created and is up and running it will then be hived off to the private sector to another firm, so there is an entire arm's length between members of Parliament and those who are dispersing amounts under allowances."
He said MPs had "lost the confidence of the public and we need to get that back".
Northern Ireland Secretary Shaun Woodward, one of the richest men in the cabinet who does not take his full ministerial salary, reportedly claimed £100,000 in mortgage interest over four years.
He told the BBC: "I think the public look at us right now, think we have all got our noses in the trough - our profession looks shameful - it's embarrassing.
"Right now if I try to make almost any defence of our collective position - or my position - it looks terrible."
He said MPs' expenses had to be looked at - but other public bodies, like the BBC, should also release their expenses details.
If the reports are in all cases correct, then there clearly are instances where MPs have lost contact with the difference between right and wrong
The Sunday Telegraph published a series of new revelations about MPs, including nearly £500,000 claimed by Sinn Fein MPs, who do not take their seats in Parliament.
Sinn Fein MP for West Tyrone Pat Doherty said they were "only claiming what we're entitled to".
"We're transparent and we're open. In fact we're the only party that publishes annual financial accounts so we have nothing to hide in all of this," he said.
It also reports that the Conservative former minister John Gummer claimed more than £9,000 a year for gardening - and £100 to have moles removed from his country estate.
Mr Gummer said he had not claimed for mortgage costs or food or furniture and had only claimed "the relevant proportion of the costs of necessary maintenance and repairs of an old rural property".
Communities Secretary Hazel Blears also faced questions about the sale of her designated second home - a London flat on which she claimed expenses - after it emerged she had not paid capital gains tax on the sale.
Properties which are considered a main residence by the Inland Revenue are not subject to 40% tax when sold. Her spokesman said there was "no liability" for the tax on the flat.
Speaking outside her Salford home, she said she had "complied with the rules of the House, the rules of the Inland Revenue" but added: "I understand entirely why the public hates this. The system is wrong, it needs to be changed."
Both she and another minister, Andy Burnham, called for the public to be involved in setting up a new, tighter system.
Mr Burnham - whose own claims have been examined by the Telegraph - told Sky News: "It can't look like a solution has been stitched up in Westminster. I think there has to be some public endorsement."
It has been revealed Ms Blears claimed for expenses on three homes in a year
Meanwhile Mr Miliband told the BBC One's Andrew Marr Show: "What we are seeing this weekend... is a challenge and a wake-up call to politicians about the systems that we have in place."
He said: "Of course it needs to be reformed. I take my responsibility as an MP that we didn't reform it earlier."
For the Liberal Democrats, Chris Huhne said: "If the reports are in all cases correct, then there clearly are instances where MPs have lost contact with the difference between right and wrong.
"I think we need to make sure we're saying that loud and clear because, frankly, the voters are not going to be at all sympathetic if we don't."
Millions of receipts backing up all MPs' expenses claims under the second homes allowance were due to be published in July after a long Freedom of Information campaign.
But details have been leaked to the Telegraph - which has been publishing selected excerpts over the last few days.
It has concentrated largely on Labour ministers, but says it will publish details of other parties' claims next week.
Conservative defence spokesman Liam Fox said every MP would have to answer for themselves.
But he told the BBC: "The trouble is that politicians have tended to say 'we were only acting within the rules' but the public think the rules themselves are wrong and go way beyond the legitimate needs of people who have to be in London to represent their constituents in Parliament."
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