According to a Department for Education survey of local authorities, 44 per cent of councils said they had experienced difficulties in providing 15 hours a week of funded childcare for disadvantaged two-year-olds.
Nearly four out of 10 councils said the doubling of the funded childcare entitlement - from 15 to 30 hours a week, for three- and four-year-olds of working families in September 2017 - had harmed efforts to ensure disadvantaged two-year-olds take up their entitlement.
A key factor is that places for three- and four-year-olds are cheaper to provide than those for disadvantaged two-year-olds, the research found.
Of the councils surveyed, 16 per cent reported they found providers are unwilling to offer funded places to eligible two-year-olds.
One council representative surveyed said: "Some challenges are emerging relating to the funding of the two-year-old free early education entitlement and providers are beginning to report that it will not be sustainable for them to continue to deliver this in the future."
Another said: "We have experienced some displacement of funded two-year-olds where settings have chosen to prioritise funded 30-hour children."
In addition, one in five said they lacked funding for publicity to effectively communicate the benefits of the entitlement.
The survey found that around a third of councils had been told by eligible parents that they did not want or need childcare for their two-year-old.
In September, a NatCen Social Research report revealed that providers find it "less financially lucrative" to offer funded places for disadvantaged two-year-olds and were instead focusing on places for three- and-four year-olds.
Analysis by the National Day Nurseries Association in November highlighted the financial pressures early years providers face. This includes having to meet the cost of funded places from their own pockets due to low funding rates from government, as well as having to meet national living wage and business rate commitments.
Next year, the average nursery faces a £28,704 shortfall to meet the cost of providing 30 hours of free childcare to three- and-four-year-olds living with working families, the NDNA estimates.
Jonathan Broadbery, NDNA head of policy and external relations, said: "Once again the government is being warned about the impact of its underfunded childcare policies but this time the message is coming from councils. They are confirming what the nursery sector has been saying for years, that failing to fund places properly is having detrimental effects across the board.
"That four in 10 councils report that the 30 hours policy is damaging the number of two-year-old funded places should set alarm bells ringing for ministers. Giving very young children from deprived areas access to high-quality early years education is the best way to reduce the attainment gap between disadvantaged children and their peers.
"What we are seeing is more and more nursery providers being forced to limit the number of places they offer because they are not paid enough to cover their costs."
Neil Leitch, chief executive of the Pre-school Learning Alliance, said: "This report makes clear that the government is failing some of our most disadvantaged children. Everyone agrees that the early years is a vital in ensuring children get the best start in life - but this report shows that children from low-income families, who stand to benefit most from childcare, can't rely on the government to ensure they're able to access it.
"The fact is that ministers were warned repeatedly that providers would struggle to deliver childcare for younger children following the rollout of 30 hours. They chose to ignore those warnings and continue to do so; this study is just the latest to underline the wider impact that frozen and falling funding levels are having, not only providers but also on the children they are able work with."