Prime Minister Gordon Brown said he would consider doing the same to Labour MPs.
The Conservatives say 14 members of the shadow cabinet have been asked to repay money, although three are querying what they say are inaccuracies.
Mr Clarke has been asked to repay the most - £4,733 mainly for gardening and cleaning - and will be "making representations", but has agreed to pay.
Others include Mr Hague, who has been asked to repay £642 relating to a mortgage payment and chief whip Patrick McLoughlin who has been asked to repay £4,058.
Mr Lansley has been asked to give back £1,728.
Mr Cameron told GMTV: "If people are asked to pay back money and if the authorities determine that money should be paid back and they don't pay it back, in my view, they can't stand as Conservative MPs. That is the minimum point.
"That is the least we can do to try and sort out these problems of the past before going on to the future."
IAIN WATSON, POLITICAL CORRESPONDENT
Opinion is split on whether legal action by disgruntled MPs would succeed. So far no-one has instructed a brief, but senior parliamentarians are suggesting some of their colleagues who are faced with repayments will take legal advice.
Jonathan Caplan QC believes MPs might have a case if they are asked to pay money back on the basis of a retrospective rewriting of the expense rules - though it is also possible that a court might judge their claims excessive.
But the former chairman of the Committee on Standards in Public Life, Sir Alistair Graham, believes the prospect of legal action is remote.
That is because Sir Thomas Legg's requests for repayments would have to be endorsed by the the House of Commons Members' Estimate Committee - chaired by the Speaker. So, in effect, disgruntled parliamentarians would have to sue Parliament - a concept Sir Alistair believes would be absurd.
However, Conservative leader David Cameron's assertion that any of his MPs who don't pay back expenses, as requested, would be prevented from standing for re-election might just land him in legal trouble.
Anthony Scrivener QC told the BBC that if MPs were deprived of their jobs over an unwillingness to pay up, this could be challenged in court.
So party leaders who talk tough could find that they, rather than Parliament, have their decisions scrutinised by m'learned friends.
Among the cabinet, Business Secretary Lord Mandelson has repaid £800 relating to gardening claims and Chancellor Alistair Darling has repaid £554 claimed for a chest of drawers.
Communities Secretary John Denham has paid back £1,500, Justice Secretary Jack Straw £600 and Northern Ireland Secretary Shaun Woodward £1,426.
Meanwhile, Schools Secretary Ed Balls has repaid £13.50 for a miscalculation on a mortgage interest payment.
Sir Thomas was asked by Downing Street to review all MPs' claims under the controversial second homes allowance since 2004, after the expenses scandal broke in May.
But he has written to some MPs asking them to pay back the difference between what they claimed, approved at the time, and what he thinks they should have claimed.
He cannot force them to repay money and his final recommendations will go to the Commons Members Estimate Committee, which will decide what to do next.
Ministers said they expected many Labour MPs to write to Sir Thomas and object to his decision to apply new limits - £1,000 a year for gardening and £2,000 a year for cleaning.
Parliamentary Labour Party chairman Tony Lloyd was also asked to approach the chairman of the Conservative backbench 1922 Committee about a joint response to the review but Labour says a meeting will not take place.
The BBC's deputy political editor James Landale said some Labour MPs were planning to challenge any requests for money while others were angry at the Parliamentary Fees Office which they say has lost some of the receipts.
EXPENSES: THE STORY SO FAR
MPs are allowed to claim expenses for running a second home but there was much uproar in May when receipts and details of what they had been claiming for were leaked to a newspaper.
Among them were claims for expensive TVs and furniture, MPs who claimed for more than one property by "flipping" the designated second home and others who over-claimed for mortgages or services.
Many MPs have announced they will be standing down, some have already repaid claims in response to constituents' anger.
Party leaders pledged to change the system and an independent review is due to make its recommendations this month.
The PM also asked an independent auditor, Sir Thomas Legg, to go over past claims again, to ensure money had been paid out properly. MPs have been getting his letters saying how much they should pay back.
Gordon Brown, who has repaid more than £12,000, largely from cleaning and gardening claims, has urged others to do the same.
He said it was "the last stage" in the "old discredited system" adding: "I would advise any MP after they have made their representation to make the appropriate payments."
Asked if he too was ready to withdraw the whip from his MPs if they refused, he said: "If, of course, people are not prepared to co-operate, then we will have to consider that action."
Tory veteran Ann Widdecombe said she thought Sir Thomas's requests were legally questionable.
She told the BBC: "If any other employer said to his employees: 'These were the rules. You stuck fastidiously by them ... but we have now changed the rules so here's a bill', that employer would be up before a tribunal."
And Labour's Martin Salter, who does not claim for a second home himself, said there was "a lot of anxiety and anger" among MPs.
The Liberal Democrats say they will not be publishing details of Sir Thomas's requests to their MPs.
Asked if he would force his MPs to pay up if they refused, leader Nick Clegg said: "I just don't think that's going to happen." He said he expected all MPs to "co-operate fully".
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