The £2.4bn schools in England get to teach pupils from disadvantaged homes should be doubled, a think tank said.
A "pupil premium" would encourage schools to take poorer pupils
The Centreforum report, Tackling Educational Inequality, said this could fund evening and weekend classes which children might be forced to attend.
It said low attainment too often stemmed from children's backgrounds, not their abilities.
The government said much support was already available and compulsory extra lessons would not motivate pupils.
"Britain is a bastion of educational inequality," said Paul Marshall, chairman of Centreforum, an independent liberal think tank.
"The die is cast at an early age and rather than recast the die, the English educational system tends steadily to reinforce the advantages of birth."
He said: "The system isn't failing the least able, it's failing the most disadvantaged."
The proposals would see funding for disadvantaged pupils rising to up to £8,000 per year, in the same bracket as many independent day school fees.
This "pupil premium" would operate like a form of voucher scheme - so not only would the funding follow the pupil, but more funding would follow the most disadvantaged pupils.
"The pupil premium would enable more resources to flow directly to schools with disadvantaged intakes and would for the first time give successful schools a clear incentive to admit pupils from deprived backgrounds."
Mr Marshall put forward a "menu" of options from which schools would be free to choose. These included:
Other ideas at little or no cost include freeing up the curriculum, having outside professionals filling a revised head teacher role, and ensuring teachers "seize the opportunities of the data revolution".
- Smaller class sizes at primary school
- Significantly more hours of teaching time for disadvantaged pupils (including new powers for schools to make longer school days compulsory, Saturday schools and summer programmes)
- 'Hard to Serve' bonuses, linked to performance related pay schemes, for staff in the most challenging schools
The Minister for Schools and Learners, Jim Knight, said the government was aiming a wide range of support and funding at schools in deprived areas and at children from disadvantaged backgrounds.
"Extended schools already offer services outside the school day. If a school wished to open on Saturdays that would be a matter for the school and its staff to decide.
"While we have introduced the option of weekend detentions for schools to use if they wish, in general we do not think that compelling children to take extra lessons at weekends is likely to engage them in learning or motivate them to succeed."