David Cameron has urged the Royal Bank of Scotland to show restraint in its bonus packages for senior colleagues of chief executive Stephen Hester, echoing an earlier call by the business secretary, Vince Cable.

Cable became the first cabinet minister to urge other bankers to show the same restraint as Hester, who bowed to pressure from the Labour party and public opinion by waiving his £1m bonus.

Downing Street repeatedly said on Monday it would not tell the board of RBS to rein in bonuses, saying it was vital the existing management team at the government-owned bank remained in place and it was not the job of the government to micromanage the company.

Cameron's spokeswoman praised the RBS board for reducing the balance sheet and making order out of the complete mess it had inherited, but claimed that the previous Labour government had left the coalition with contractual obligations allowing the board to pay bonuses linked to performance.

But , at an EU summit in Brussels, Cameron said: "They have got to have proper regard in terms of restraint when they have had so much money from the taxpayer and they have made so many mistakes in the past." He added at a press conference: "They need to do a better job – as everybody has – about explaining how pay is linked to performance."

Cable had earlier suggested other bankers should follow Hester's example. "I think it's a good step to dealing with the bonus culture more generally. I'm not dictating … but I think he has set a good example. I think that more generally in the banking sector we have got to see more of a sense of perspective."

His remarks suggest that, as the bonus season approaches, some coalition cabinet members are likely to criticise the packages of individual bankers, fearing that the government's claims to be cracking down on crony capitalism will look hollow if they do not.

Among the next public targets will be Bob Diamond, the chief executive of Barclays, who is in line to receive up to £10m, and John Hourican, head of RBS's troubled investment banking division, who has just announced 3,500 job losses. He is due to receive a £4.3m windfall in April.

The prime minister had privately suggested that Hester should take a bonus below the headline sensitive £1m figure, but did not feel it was right to press for a bigger cut.

There was some frustration in Liberal Democrat circles that the Treasury and UK Financial Investments, which manages the government's shareholding in RBS, told them they had no option but to let Hester receive the payout. Some Lib Dems were told Hester might walk away from the job if too much pressure was applied and this could lead him to suing the government, ending up with the episode becoming even more expensive.

Hester's aides said he decided to forgo his bonus at the weekend rather than become singled out for criticism in a Labour-led Commons debate set for next Tuesday.

Ed Miliband said he intended to step up the pressure for change in UK boardrooms: "This cannot be a one-off episode. There has to be a change in the culture. We have got to learn the deeper lessons. We have got to have responsibility in the boardroom and that is why I say tax the boardroom bonuses and give ordinary workers a say in their boardroom on pay."

The Labour leader moved to capitalise on the episode by announcing he would table a Commons motion calling on the government to tax bonuses until bankers start to show restraint. He is urging Conservatives and Lib Dems to back a £2bn banker tax, which Labour says could fund the creation of 100,000 jobs elsewhere.

A senior party source said: "This debate is about more than Stephen Hester. It is time for the Conservatives and Lib Dems to show they are serious about tackling the bonus culture by supporting our plan to repeat the bank bonus tax."

Miliband accused Cameron of having a tin ear for public opinion. The Labour leader's aides will be delighted at his strong lead on the issue, a reminder of his tough stance over the need for an inquiry into News International.His strategy suggests Labour will, at least in the short term, focus on systemic reform rather than target the bonuses of individual bankers or the specific remuneration of the rest of the RBS board.

Labour officials said they had not yet seen the details of other bonuses to be announced in the near future.

Miliband was praised for his lead by the former City minister Lord Myners, who nevertheless warned Labour not to seek scalps or put itself in a position where it appears hostile to business.

He told Miliband not to allow his campaign to tip over into anti-business sentiment or scalp-hunting.Some senior Labour figures believe popular opinion cannot be assuaged on the issue of bankers bonuses and it may prove necessary for Labour to set out clear principles.

The shadow business secretary, Chuka Umunna, described RBS investment managers as public sector workers and said his party would take a close look at what other RBS star bankers such as Hourican received.

Myners said: "At some point, somebody has got to say 'look this has just gone too far'. This is unacceptable. Royal Bank of Scotland is different from other banks because it's publicly owned. It may be a private company but it's 82% owned by the taxpayer. And this problem will keep coming around year after year until the government has ceased to have a controlling interest.

"The board of directors can quite simply say 'we can no longer justify these very high rewards for bankers'. It's not as profitable as it used to be, we're slimming the bank down, and therefore we have to be much more realistic in terms of what we pay. I think that should apply across the whole banking sector."