(Read more: The battle to be the centre of renminbi trading)
Any agreement will not be ready in time for Prime Minister David Cameron's trip to China this week,but could be announced early next year, according to one source close to the Chinese, who added that there was no certainty a deal would be made.
(Read more: China charm offensive raises eyebrows in UK)
The U.K. government still owns 82 percent of RBS and 32.7 percent of Lloyds' shares, after bailing out both banks during the credit crisis. It sold off part of its Lloyds stake earlier this year, making a £61 million ($99.6 million) profit. Meanwhile, RBS's share price has fallen by around 9 percent this month after announcing that it will create an internal "bad bank" of its riskiest assets.
"When these stakes are sold, our responsibility is to get the best price for them," a source with knowledge of U.K. government thinking told CNBC.
The U.K.-China thaw is partly because there is now more Chinese money to be invested internationally, after a pick-up in credit following a mini-stimulus program announced by the Beijing government in July. Plus, the U.K. economy is starting to pick up, with London's safe haven status and the relatively weak pound helping to stimulate investment.
Mike Trippitt, banks analyst at Numis Securities, said that any investor able to take a two-to-three year view of the primarily domestic U.K. banks would be "extremely interested" in them.
"There's definitely value in RBS shares. Lloyds also have a bit of upside, but the valuation is reasonably full. Barclays' gross leverage is still an issue," he told CNBC.
(Read more: UK banking bonuses up again)
U.K. Financial Investments, the body which manages the government stake sale for Lloyds and RBS, declined to comment, as did the Treasury.
Chinese investment in the U.K. dropped off after Cameron's well-publicized meeting with the Dalai Lama last year. However, a series of Chinese-U.K. deals have been announced in the past couple of months, despite concerns about previous investments. These include ICBC's investment in a £650 million ($1 billion) business park near Manchester Airport and a high-profile investment by China General Nuclear Power Group in a new nuclear power station in southwest England.
(Read more: Yuan moves one step closer to global currency status)
A Barclays spokeswoman told CNBC that the bank would not comment on speculation. The Co-op and Lloyds also declined to comment and RBS and Norton Rose Fulbright had not returned calls by the time of publication.
- By CNBC's Catherine Boyle. Twitter: @cboylecnbc.