The prime minister confirmed that the government was examining an "interesting" idea that would involve shares in the bank initially distributed to taxpayers for free. The Treasury would make a profit once the shares reached a certain price.
"These are all interesting questions for the future," Cameron said of a variation of an idea floated by the Liberal Democrats in 2011, and recently picked up by the Treasury minister, Sajid Javid.
On Saturday, the Daily Mail and the Independent reported that Javid has been told by Chancellor George Osborne to draft plans for what was described as a "give-away" of RBS shares. This would help the government recoup a decent chunk of its £45bn bailout of the bank.
Cameron said "I am keen to examine all possibilities for what we could do to put RBS in time back into the private sector. I can think of lots of exciting and interesting things to do but I don't have a fixed view."
But the prime minister, who is in India, said his main focus was to ensure the bank was in a fit state to be sold off. "The first job is to turn around the performance of RBS and to strengthen its balance sheet, strengthen its business and that's what [the chief executive] Stephen Hester and his team are doing."
Cameron declined to specify what he meant by "in time", but made clear that the government's patience in RBS was not limitless.
"It's a huge undertaking, it was a very badly damaged institution," he said of the bank. "But I think they are doing the right thing and obviously we want them to, where possible, accelerate the adjustments that they are making in terms of making it a strong organisation."
The prime minister's interest in the public share option follows Vince Cable's speech earlier in February in which the business secretary outlined options for the future of RBS.
Cable said: "There is a range of options, from reprivatisation at a later stage to continued public ownership or mutualisation through public share distribution, as advocated by the Liberal Democrats. We should keep all of these options in play."
But he later played down the idea telling Radio 4's Today programme that there was "no immediate prospect" of handing RBS shares over to the public.
Fellow Lib Dem MP Stephen Williams was the driving force behind a plan drawn up by Portman Capital Partners to give RBS and Lloyds Banking Group shares to the 46 million adults registered to vote in the UK. A fixed nominal amount would have to be paid to the Treasury when the shares are sold, ensuring the government would recover the £66bn bailout of UK banks.
Government sources stressed that Cameron was adopting a cautious approach on RBS. On the one hand he is keen to ensure the taxpayer is compensated for the investment it has made, while on the other he is aware it would be wrong and counter-productive to act precipitously.
"It is better to wait and get it right than to rush and get it wrong," one source said.
Other sources were more encouraging about the share plan. One senior figure told the Independent: "There is a realisation that there is no prospect of RBS's share price rising to the level at which we bailed the bank out and it's not good for the bank or the government to hold on to our stake indefinitely. Obviously a give-away to taxpayers before the election, who after all paid for it in the first place, is very attractive."
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