RBS is seeking to sell between 25% and 33% of the business and set a price range of 160p to 195p last week.
"The book is now fully covered within the range, and to get that done three days in is pretty encouraging," a source said yesterday.
Direct Line would be worth nearly £2.7m at the mid-point of that range.
If priced at 195p, RBS could raise as much as £975m, which would also make the sale London's biggest initial public offering in more than a year.
The apparent success comes despite the Office of Fair Trading decision to ask the Competition Commission to investigate the car insurance market.
This has prompted fears the sector could become much less lucrative for insurers in future.
It was reported that RBS's bankers have so far focused on just UK investors and that further demand could be drummed up in North America and continental Europe.
RBS is being forced to sell Direct Line, which also operates brands including Churchill and Green Flag, as part of the conditions for European Union regulatory approval of its £45 billion state bailout during the 2008 financial crisis that left it 82% state-owned.
RBS must sell a majority stake in Direct Line Group by the end of next year, and divest of the entire company by the end of 2014.
Order books for the initial public offering, which opened on September 28, are due to close on October 10.
The Edinburgh-based bank said it expected to set final pricing on or around October 11.