Downing Street insisted the Government was not going to "micro-manage" the part taxpayer-owned RBS, after the waiving of a £1 million bonus by Stephen Hester, its chief executive.
With John Hourican, head of the bank's investment arm, said to be in line for a £4.5m shares bonus in April, the Prime Minister's spokeswoman said: "These decisions are decisions for the board, in terms of who gets a bonus and what they get. The Prime Minister has made very clear that bonuses should be responsible."
She said Mr Hester had made good progress in turning round the Edinburgh-based bank, which was saved from collapse in 2008 by a massive bail-out and had previously been "a complete and utter mess".
The spokeswoman added that rewards were a "matter for individuals" on whether to accept them or not. It follows claims by Vince Cable, the Business Secretary, that Mr Hester's waiving of his £963,000 bonus, following on from RBS chairman Sir Philip Hampton's decision to give up his £1.4m reward, had "set the tone".
The Prime Minister's views emerged as Labour leader Ed Miliband went on the offensive over the issue, insisting the time had come to tackle Britain's bonus culture.
Mr Miliband, speaking in Glasgow, said Mr Hester's decision – which he announced on Sunday, two days after his payout was revealed – could not be just a "one-off episode".
Mr Hourican, 41, who will oversee a restructuring of the 82% taxpayer- owned bank, including around 3500 job losses, is due to pocket £4.5m in long-term incentive shares awarded to him in 2009 on condition targets were met.
The Labour leader accused the UK Government of nodding through Mr Hester's bonus and showing a "tin ear" when it came to hearing and understanding public concerns.
On the subject of Mr Hester, he added: "I don't think this can be just a one-off episode because if we don't deal with this systematically, if we don't deal with the issue of bankers' bonuses in a proper way, this kind of thing is just going to re-occur."
At Westminster, it is thought Nick Clegg, the Deputy Prime Minister, is unhappy that the Treasury has not put more pressure on the RBS board, which approves the bonuses.
Mr Cable last night made clear that Mr Hester's decision had "set a good example", adding the banking sector had to show "more of a sense of perspective".
Labour, which believes it has shown the Conservatives to be out of touch with public sentiment on the bank bonus row, will next Tuesday use a Commons debate to urge Tory and LibDem MPs to back an extra £2 billion tax on bankers' bonuses, which they argue would fund the creation of 100,000 jobs elsewhere.
"This debate is about more than Stephen Hester," declared a senior party source.
"It is time for the Conservatives and LibDems to show they are serious about tackling the bonus culture by supporting our plan to repeat the bank bonus tax."
Earlier, ministers welcomed Mr Hester's decision to waive his bonus. George Osborne, the Chancellor, described it as a "sensible" decision, which would enable the bank chief to focus on recovering the £45bn the taxpayer ploughed into RBS to save it from collapse in 2008.
That year, Mr Hester replaced Sir Fred Goodwin as chief executive.
His contract stipulated he should be considered for a bonus based on performance against a set of goals, which included increasing lending and reducing the bank's £2 trillion balance sheet.
Although RBS's share price has tumbled by more than one-third over the past year, No 10 pointed out £53bn had been lent to businesses and £500bn had been wiped off the bank's balance sheet.
David Fleming, of the Unite union, claimed it was "common sense" for bosses to waive bonuses at a time when RBS was imposing large numbers of redundancies on frontline staff.
"This gesture goes some way in acknowledging the hypocrisy of an organisation which has sacked over 21,000 staff while still attempting to pay bumper bonuses to the bosses," said Mr Fleming.
"There remains a long way for RBS to go in proving its credentials as a responsible organisation, to its customers and also to its thousands of staff," he added.
Elsewhere, Hector Sants, head of the Financial Services Authority, told MPs none of the executives running RBS at the time of its near-collapse would get top jobs in financial services again.
His remarks close the door on a return to banking for Sir Fred and his former colleagues at the Edinburgh-based bank, even though an FSA investigation concluded there was no case for enforcement action against them.
"In my personal judgment, the set of failings demonstrated over a prolonged period there is a clear track record of a series of misjudgments by the executives of RBS, and that means they are not fit to run a regulated institution," Mr Sants told the Commons Treasury Committee.
He added: "As far as I am concerned, if any one of those individuals was to apply, I would not consider them competent under the track record demonstrated."