David Cameron has rejected calls to soften the impact of planned cuts to tax credits in the face of a cross-party campaign that warns that 3 million of the poorest working families will be worse off by £1,350 a year.

In an interview on the Andrew Marr Show on BBC1, in which he said the Conservatives would champion the “common ground” as Labour “runs to the hills”, the prime minister defended the cuts on the grounds that the government has to protect the NHS and schools.

Facebook Twitter Pinterest Frank Field. Photograph: Anthony Devlin/PA

Cameron indicated that the chancellor, George Osborne, would not revise his plans to cut tax credits when he delivers his autumn statement next month, despite warnings from former Tory minister David Willetts and the Labour welfare expert Frank Field. The Sun on Sunday launched a campaign to help low-paid workers under the headline “Tax credit cut ‘bonkers’”.

Asked whether the chancellor would have a rethink in the autumn statement, the prime minister said: “No. We think the changes we have put forward are right and they come with higher pay and lower taxes.”

Cameron made his remarks on the first day of the Conservative party conference in Manchester, in which he also:

• Teased Osborne, seen as the frontrunner to succeed him as Tory leader when he steps down by 2020, by saying that frontrunners often do not win.

• Hinted that cabinet ministers might be allowed to campaign on either side in the EU referendum, even as he indicated that he was determined to remain in the EU.

• Illustrated the Tories’ plans to occupy what he called the “common ground” of British politics, as Labour moves to the left under Jeremy Corbyn’s leadership, by announcing that GPs’ contracts were to be rewritten to ensure the NHS becomes a true seven-day service.

Moderate Tories who have welcomed the prime minister’s move to the centre ground have warned that the cuts to tax credits, outlined by the chancellor in his budget in July, could undermine Downing Street’s attempts to appear as the “workers’ party”. Lord Willetts – one of the most influential Tory thinkers of the past decade, who served as science and universities minister for most of the last parliament – warned in the Times last week that the changes could “turn sour”.

The chancellor outlined the cuts in his summer budget when he announced that he would slash the earnings level at which tax credits start to be withdrawn from £6,420 to £3,850 from next April. Osborne said that low earners would be compensated by the creation of a “national living wage” – a rebranding of the minimum wage – which would be set at £7.20 an hour from next April for over-25s. It is to rise to £9 an hour by 2020.

The Institute for Fiscal Studies said in its post-budget briefing that 13 million families will lose an average of £240 a year, while 3 million families will lose £1,000 a year. Paul Johnson, the IFS director, said it was “arithmetically impossible” for the increase in the minimum wage to compensate for the loss in tax credits.

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The prime minister insisted that families would not be worse off. “If you take a family where someone is on the minimum wage, when you take into account all the things we are changing in tax, in the national living wage and tax credits – that family will be better of, not least because you’ve got the national living wage, and we are cutting tax so you can earn £11,000 before you start paying any tax at all.”

But Cameron said he would look at a proposal by Field, the Labour chairman of the Commons work and pensions select committee, who has proposed a revenue-neutral way of reducing the impact of the tax credit cuts on the lowest paid. Field says that the current income threshold for the withdrawal of tax credits should remain at £6,420. This would be paid for by withdrawing tax credits more sharply – a taper rate of 65% – on gross earnings over £13,100.

Cameron said of the Field proposal: “I always look at everything. But we have had the vote in parliament on tax credits. And people do respect this argument that the national living wage, a 50p increase next year, so a £20 a week pay rise, rising to £9 by the end of this parliament – that is a very significant change that really helps to make work pay rather than a tax credit system that recycles money back to people. Obviously, I accept we are making changes to tax credits. We are protecting the lowest-paid people with the child tax credit and what goes with it. But we are moving to an economy where you get paid more and where you pay less in tax, rather than paying more in tax and getting the money back in tax credits.”

Ruth Davidson, the Conservative leader in Scotland, told a fringe event that she would like to see more detail on how tax credit cuts will work in practice.

“I agree with the prime minister that we should be absolutely cleaving to the principle that work should pay in and of itself and we should be moving to a higher wage and lower welfare economy...

“But when it comes to the application of the policy, I think we do need to see a little bit more detail about how it will work in practice and I would expect that to be brought forward in the autumn statement.”

She also said she would like to see carer’s allowance brought into line with jobseeker’s allowance, and will campaign for this to happen in Scotland once the welfare budget is devolved.