Stephen Hester will be given about £780,000 in shares next month, as part of a reward scheme for his performance in 2010.
Details of the bonus come shortly after RBS agreed to pay £391m to the British financial watchdog after admitting staff manipulated the Libor rate, a key benchmark interest rate on which trillions of pounds worth of deals are based.
Lord Oakeshott, the Liberal Democrat peer, said: "It is wholly unacceptable that Hester should receive a bonus for 2010 when these scandals were still going on.
"He had been captain of the ship for two years, but the crew was still robbing the passengers."
However, Mr Hester vowed that he would stay to "finish the job" at the bank despite damning evidence from US and UK authorities over its role in the Libor scandal, dating back to 2006 and continuing through to late 2010 – when investigations had already begun.
Mr Hester will be handed his bonus in shares next month, and will be able to cash them in 12 months later.
The exact value will depend on the share price when he cashes them in.
Mr Hester's payout next month will be the second tranche of a two-part reward scheme first announced in 2011. RBS, which is 81% owned by the Government, will recoup about £300m from its staff bonus pool and by clawing back previous awards to pay for the fines.
John Hourican, head of RBS's "casino" banking wing, has left the firm and been forced to forego £4m of bonuses awarded in previous years. However, the bank has decided not to take back Mr Hester's award, even though it coincides with the period when Libor rigging was happening.
RBS chairman Sir Philip Hampton said: "We are not contemplating, for the avoidance of doubt, clawing back Stephen's bonus award that was made in 2010.
"Stephen has only had one bonus out of four years, which we think is already quite a severe level of clawback through a different route."
Mr Hester replaced the disgraced Fred Goodwin as chief executive of the RBS Group in November 2008, and is paid an annual salary of £1.2m.
In 2010 he took home £6.5m in bonus and pension payments.
He went on to decline bonuses in subsequent years due to political pressure and in light of an RBS computer glitch that disrupted the accounts of thousands of customers.
RBS has also reportedly asked former directors – including Fred Goodwin – to return bonuses "earned" while the Libor rigging and mis-selling of payment protection insurance were continuing, but none has agreed to do so.
Mr Goodwin, who was stripped of his knighthood in the wake of the scandal, accumulated about £6m in bonuses during the period when rate rigging was going on.