U.S. buyout giants Blackstone Group (BX) and Bain Capital (BCI.XX) are assembling a pre-emptive bid due at the end of the month that would derail the flotation, which is seen in September.
Another group, comprising Kohlberg Kravis Roberts, Apax, and BC Partners, is also putting together a rival offer, the newspaper added.
Direct Line is the U.K.'s biggest car insurer by number of policies, and a leading provider of home insurance. Its brands include Churchill and Green Flag, the car breakdown service, the newspaper said.
European regulators ordered RBS to sell its insurance arm as a condition of a government bailout four years ago. RBS, which is 84% owned by the British taxpayer, received GBP45 billion to save it from collapse, the newspaper said.