The National Day Nurseries Association (NDNA) survey found widespread concern among providers over the hourly funding rates on offer for free childcare, which will double to 30 hours per week for three- to four-year-olds from September.
Of more than 1,000 providers surveyed in July, 15 per cent said the rate was so low that they would not offer the additional hours.
Among the 85 per cent that have agreed to offer the entitlement to parents, 60 per cent said this would only be possible by charging parents extra for meals and activities.
Half of those offering the additional hours have said they would restrict the number of places on offer, in some cases to only one or two children.
"There is a real danger that there simply won't be enough places to meet demand and thousands of parents will end up disappointed," said NDNA chief executive Purnima Tanuku.
"We have lobbied the government hard since 30 hours was first committed to in May 2015 to make sure sufficient funding was put in place to make this a success. This has simply not happened.
"Average increases were just 40p per hour from previous research, so now nurseries are put in an extremely difficult position. If they go ahead with the scheme, they risk their sustainability. If they don't, they risk losing business as parents vote with their feet."
She added that while additional charges are allowed, latest Department for Education guidance states they must not be a condition of accessing a place and providers must offer alternatives to parents who do not wish to pay.
In the North West of England and outer London the proportion of nurseries that will decline to offer the additional hours is above the national average, at 23 per cent and 19 per cent respectively.
The areas with the highest proportion of nurseries set to offer the additional hours are the East Midlands (91 per cent), East of England (89 per cent) and the North East (88 per cent).
Most nurseries told the NDNA that they felt they had no choice but to offer 30 hours despite funding concerns.
"For me to take part, the government needs to change the word 'free' to 'subsided', said Carol Medcalf, owner of Carol Jane Montessori Nursery School in Enfield.
"I then can't think of a reason why any early years setting would not take part. The government would be advertising the 'reality', parents would know what to expect and nurseries could charge according to their cost. Children would receive the quality they deserve."
Last month, a government-commissioned evaluation found that nearly half of private childcare providers that have already piloted the 30 hours' free childcare offer, have reported a dip in profits.
Minister for children and families Robert Goodwill said: "Across the country we are already delivering the 30-hour offer with great success - over 15,000 children are benefitting from a place and our evaluation shows that providers are committed to offering this to parents.
"In York, one of the 12 areas which has been piloting the scheme since last September, 100 per cent of nurseries offering the original 15 hours are now also offering 30 hours. In Northumberland, 100 per cent of providers have signed up to deliver the 30 hours. We are also hearing similar results from the other early implementer areas.
"We have always been clear that our 30 hours funding is expected to cover the cost of delivering free childcare. If providers need to charge for extras, such as meals or activities, this cannot be a condition of a child accessing their place."