The chairman of Royal Bank of Scotland (RBS), Sir Philip Hampton, has said the bank will be ready to return to the private sector next year.
In a video statement posted on the bank's website, Sir Philip said he expected the government to start selling shares from the middle of 2014.
His comments came as RBS reported a return to profit for the first three months of the year.
It made a pre-tax profit of £826m after racking up losses last year.
RBS lost £1.5bn in the first quarter of 2012 and lost £2.2bn in the final quarter.
But investors reacted negatively to the results. RBS shares fell more than 4.5% in the first 10 minutes of trading on the London Stock Exchange.
Clean up 'almost complete'
Sir Philip described the results as "our best quarterly performance for some time now", opening the door for a return to the private sector.
"Our balance sheet is substantially fixed... our operating profitability has come through quite strongly," he said.
"What we want to do is have a business that is performing well... enabling the government to start selling shares from, let's say, the middle of 2014 on - it could be earlier, that's a matter for the government - but certainly we think the recovery process will be substantially complete in about a year or so's time."
That could mean a return for shareholders after the government invested tens of billions in the bank to rescue it from collapse in 2008.
Speaking to the BBC's Today programme, RBS chief executive Stephen Hester said RBS will be looking "much more like a normal bank" next year, when the "cleanup will be substantially complete".
"We can deliver an RBS than can do its job, and is cleaned up, in the not-too-distant future," he said, but added that the sale of the bank would be a decision for the government.
He also said he was "open" to the idea of dividing the bank into a good and bad bank to rid itself of bad assets - a plan backed by the Bank of England governor, Sir Mervyn King.
But speaking later on a conference call with reporters, Mr Hester said he had not held any explicit talks with the government about a possible sale.
It is unclear at what price the government would be prepared to sell its stake in RBS back to private investors.
The government bought shares as part of its bailout at 502 pence a share, but they are valued at 407 pence a share on the government's accounts.
That suggests the Chancellor, George Osborne, could opt to sell at the lower price and still claim to be getting fair value for the 82% taxpayer stake.
Mr Hester allowed for the possibility that the government might consider selling shares at a loss initially, but said the average sale price "can and should be" above the government's 502 pence purchase price.
Currently, shares are below 300 pence.
The share price fell further when markets opened on Friday morning, with analysts unconvinced by the results, arguing that the headline figures disguised a weaker underlying performance.
RBS said its losses for the quarter relating to bad loans were down 26% to £1bn, and it had now seen a 79% reduction in non-core assets since it began restructuring in the wake of the financial crisis.
It also reported a modest rise in lending to businesses, totalling £13.2bn in loans in the quarter. It said £7.8bn was to small and medium-sized enterprises (SMEs).
But operating profit actually fell by 28% to £829m - significantly lower than some analysts had forecast.
There was a particularly steep decline in the profitability of the investment banking division, which is being wound down.
"Overall, we expect the market to be disappointed, but perhaps not too surprised, by the weak markets performance in the core business," said Gary Greenwood, an analyst at the investment firm Shore Capital.
"Overall, we remain cautious on RBS shares, which remains our least favoured stock in the sector."