The shares were originally priced between 260p to 330p, but the large amount of interest in a stake in the postal firm will mean that they are now likely to be sold for between 300p and 330p each.
It is understood bankers handling the sale have held discussions with institutional investors how the shares will be priced.
The deal to privatise the near 500-year-old company is regarded by a number of business analysts as the most significant in the UK since John Major’s Conservative government sold the railways in the 1990s.
And commentators say it would give Royal Mail access to the private capital it says it needs to modernise and compete in a thriving parcels market.
Potential investors had placed orders for all the shares within hours of the company’s launch last week.
The government, which is selling up to 62 per cent of the business and will keep the rest, has said around 30 per cent of the shares on offer are expected to go to individual members of the public, who must spend a minimum of £750 to invest in the company.
The flotation will be the largest Initial Public Offering (IPO) on the UK stock market this year.
A further 10 per cent is to be handed to staff in the largest share giveaway of any major UK privatisation. If distributed equally among the eligible 150,000 United Kingdom-based workers, each could receive £2,200 worth.
Order books are due to close on Tuesday, with the shares beginning conditional trading in London on 11 October.
However, the sale has been criticised by the Labour Party and unions, who are balloting for strike action.
Shadow business secretary Chuka Umunna expressed concern over the possible sale of Royal Mail sites after the privatisation.
He warned of fears that once the privatisation is complete these assets – many in prime locations – will be sold, giving a large windfall to investors, while the taxpayer is “short-changed”.
He also warned that customers could be left having to trek miles to inconveniently sited delivery offices.
“Royal Mail has a huge property portfolio in prime development sites in London and across Britain and there is nothing to stop the privatised company making a quick buck by flogging off these assets for development,” he said.
In the official prospectus for the share issuing, the Royal Mail highlights three sites in London at Mount Pleasant, Nine Elms and Paddington as being “surplus”, with some reports saying they were worth between £500 million and £1bn.
The Royal Mail also warned in the prospectus that industrial action posed a potential risk for the share price.
The statement said: “National industrial action, or the threat of national industrial action… could also have a material adverse effect on the success of the offer and, if it occurs after admission, could cause the price of ordinary shares to fall significantly.”
Members of the Communication Workers Union are currently voting on whether to strike over issues linked to the privatisation, although it is believed that the ballot will close after the Royal Mail is due to make its market debut.