Over one-third of those questioned by Alliance & Leicester Commercial Bank said they will find it difficult to offer the new rights when they are introduced.
Proposals to extend the right to request flexible working to parents with children under the age of 16 could become law by April next year.
But 86 per cent of small employers believe the changes are only manageable for larger firms, with 65 per cent claiming they do not have enough staff to provide flexible cover and 54 per cent insisting they will not be able to meet customer expectations.
Steve Jennings, director of business banking at Alliance & Leicester, said: "Already under increased pressure to cut costs and increase productivity amid uncertain times, many small businesses are worried about how they will manage this added burden."
Last month, the government launched a consultation to examine how dealing with flexible working requests from employees can be made easier for businesses.
