Report says bank wants second-round bids tabled by 31 July.
RBS has set a deadline of the end of this month for bids for its £7bn GWP insurance division, reports have said.
Reuters claimed that the bank, which is understood to have received first-round bids from four suitors last month, could subsequently enter exclusivity with a preferred bidder, provided the price offered and the deliverability of financing is adequate.
RBSI, which includes Direct Line, Churchill and broker-only insurer NIG, has been valued at as much as £8bn, but analysts have said it is unlikely fetch more than £6bn given difficulties in the credit markets and the poor performance of the UK private motor market, of which RBSI controls a third.
Sources have also questioned the availability of sufficient capital for the deal after RBS rebuffed the interest of private equity suitors in May, and a number of leading interested trade buyers, including Berkshire Hathaway, AIG and Generali withdrew their interest.
Reports claimed that Berkshire Hathaway had seen a provisional bid of £6bn rejected by the bank. AIG, which has been repeatedly linked with NIG, was forced to withdraw its interest after being hit by sub-prime related writedowns of over $30bn.
The remaining companies in the frame are Zurich, Allianz and U.S giants Travelers and Allstate. Allstate recently appointed Lehman brothers to advise on a potential pruchase of the business.
Three weeks ago RBS chief executive Sir Fred Goodwin expressed optimism that the business would fetch its full anticipated price, but reiterated the possiblity that the bank could opt to hold onto the business.
At the same time James Schiro, global chief of leading suitor Zurich, hinted that he would not be willing to pay over the odds for the business after the company's advisors, Citigroup, said that any price over £5.8bn would impact adversely on the company's projected return on equity targets, while questioning the ability of Zurich to extract sufficient synergies from any deal.
RBS was the worst performing banking stock on the stock exchange yesterday, falling 5.1 per cent to 204p amid fears that asset writedowns and loan defaults could increase as the economy teeters on the brink of recession.