Mr Redwood believes the cuts should be targeted at reversing the introduction of VAT on fuel, and raising tax allowances to take more low paid out of tax altogether.
"If the current public spending plans remain, there cannot be any tax cuts," Mr Redwood said on Channel Four television.
The call by the former Cabinet minister, who challenged John Major for the leadership of the Conservative Party a year ago, will be reinforced by leaders of the right-wing 92 Group of Tory MPs who are planning to meet the Chancellor next week.
The pressure for substantial tax cuts will be increased by a Commons written answer showing the tax burden has gone up under the Tories. William Waldegrave, the Chief Secretary to the Treasury, confirmed the Budget estimate that total taxes and national insurance contributions had risen as a percentage of gross domestic product from 35.5 per cent in 1979-80 to 36 per cent in 1995-6.
The Tory right-wing intends to warn Mr Clarke that unless he is prepared to find the room for substantial tax cuts, the Tories' election chances will be dashed.
Tax cuts will mean substantial reductions in public expenditure, which could also prove unpopular. Mr Clarke has said he would not "slash and burn" to achieve tax cuts, and he has emphasised he is determined to follow a "sensible approach".Reuse content