The idea was supported by 15% of respondents - with 58% demanding a lower cap.
Sponsors of the bill had wanted a lower cap — 50percentt — but their colleagues in the House would go no lower.
Harriet Harman, interim Labour leader, told the Commons on Wednesday she was "sympathetic" to a lower cap level.
Government officials told the News Corp Sunday tabloids the prime minister had agreed to consider a new means test, and a lower cap on the current maximum payout.
The chief executive of the Child Poverty Action Group, Alison Garnham, said: "A lower cap that cuts family budgets will do nothing to help parents who want to work.
Trade groups for retailers, which want a lower cap, and banks, which want a higher one, are standing by to weigh in.
Hunt also announced a lower cap on costs for people who develop care needs before retirement age, as well as free care for those who have needs when they turn 18.
Other measures could be a lower cap on the amount individuals can contribute to their super at the subsidised tax rate, changes to capital gains tax for superannuation funds, changes to tax deductions and some increase in tobacco excise.
She said the party would back George Osborne's decision to limit child tax credit to the first two children, as well as to bring in a lower cap on the total benefit entitlement of a household.
The reforms will also include free care given to those who turn 18 with eligible care needs, and a lower cap for people of working age who develop care needs before retirement age.
Having cowed Labour and the Liberal Democrats into backing him thus far, his likely game involves promising a lower cap on the strength of fresh benefit cuts in the Conservative manifesto – and then defying his rivals to make the case for a higher cap if they dare to disagree.
