A consortium led by Corsair has agreed to pay £600m for the branches, which serve around 1.8m retail customers.
RBS was ordered to sell the branches in exchange for receiving a government bailout in 2008.
Earlier this month it was reported that Corsair was in pole position to buy the branches. At the time, RBS was still considering rival bids from a Blackstone-led consortium and from W&G Investments.
In August W&G said that its proposed acquisition could be delayed beyond its two year schedule.
RBS was due to sell the branches to Santander for £1.65bn, but the deal fell through last year.
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