The bank was forced by the EU to begin a sale of RBS Insurance (RBSI), which trades as Direct Line and Churchill, after receiving a taxpayer bailout in 2008.
CVC is believed to have made the approach in recent weeks.
The insurance division is valued at £4.3bn, according to RBS’ books, and could be sold in a trade sale or through a flotation in the second half of next year.
The bank tried to sell the unit in 2008 but initially decided to exclude private equity firms from the process, even though they could be drawn to motor insurance for its high cash-generating potential.
The lender eventually relented in 2008 and entered talks with buyout firms but rejected a joint offer from CVC and Swiss Re to buy a majority stake. The sale was cancelled in 2009.
RBS and CVC declined to comment.