Almost one in 10 people earning more than £10 million a year is paying less than the 20% basic rate of income tax, the Treasury said, as it launched a defence of plans for a crackdown.

It said figures released for the first time underlined the need for action to prevent the super-rich exploiting the system of reliefs to reduce their tax bill below that of low-paid workers.

But the move looked set to do little to calm anger over one part of that effort - with evidence of a widespread backbench revolt over capping the tax breaks on charitable giving.

A poll found almost two thirds of Conservative and Liberal Democrat MPs are opposed to the measure announced in last month's budget and which has been condemned by philanthropists, good causes and senior Tories. Of 71 asked by ComRes on behalf of the Charities Aid Foundation, 46 agreed that charity donations should not be subject to the new limit with 15 supporting the policy and 10 not expressing a view.

As the latest row to surround a measure announced by George Osborne in March gathered pace at the weekend with the intervention of Tory treasurer Lord Fink, the Government pledged to find a solution. The senior peer, himself one of the most generous supporters of charity in the UK, said the change would inevitably put off other wealthy philanthropists from giving as much.

Foreign Secretary William Hague led the defence, insisting the cap was designed to fix a genuine problem of top earners exploiting the system to pay almost no income tax. Amid reports that the Treasury had drawn up alternatives however, Mr Hague repeated an assurance from the Prime Minister that the concerns would be considered "very sympathetically".

The Treasury figures showed that 6% of £10 million-plus earners paid less than 10% in tax and another 3% came in below the basic 20% rate. Fewer than three quarters paid more than 40%.

Exchequer Secretary David Gauke said the overall cap on tax reliefs was expected to bring in £300 million of additional revenue, of which between £50 million and £100 million would come from the charities cap.

He said: "We don't think it is entirely fair that the tax system, as currently designed, does mean that there are some very wealthy individuals who are essentially able to take themselves out of the income tax system."

Mr Gauke said the Government accepted that the "vast majority" of charitable giving was "not abusive in terms of dodgy charities".