Finance minister George Osborne, a possible successor to Cameron and his de facto deputy, wants to cut tax credits as part of 12 billion pounds (17 billion euros, USD 18 billion) in welfare cuts to help reduce Britain's deficit.
Opponents and many in Cameron and Osborne's centre-right Conservative party argue that the move is unfair and will hurt several million families.
The House of Lords will hold a series of votes on the issue Monday in which the government could face defeat after peers tabled motions to try and stop the cuts.
Education Secretary Nicky Morgan, an Osborne ally, suggested to the BBC Sunday that there could be some changes to the policy as media reports suggested the government could U-turn on the issue in a budget statement in November.
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Morgan said Osborne was "in listening mode" but added: "The policy is not going to change... Not the main policy".
John McDonnell, finance spokesman for the main opposition Labour party who is close to its leftist leader Jeremy Corbyn, told the BBC it would "not make political capital" out of any change.
Osborne argues that the existing system of tax credits -- in which the state tops up the incomes of people on low salaries -- is too expensive, costing 30 pounds billion a year.
He has pledged a new "national living wage" of 9 pounds an hour by 2020 and is lifting income tax thresholds to try and boost incomes.
"What are the alternatives to trimming an unsustainable welfare bill? We would have to borrow more money we don't have and burden our children with still higher debt," the finance minister wrote in a Daily Telegraph article this month.
And 63 members of the House of Lords including two former archbishops of Canterbury wrote to the Observer newspaper calling for a change of tack.