The national overspend is three times as high as last year and the report, by the Association of Directors of Social Services, said the most significant pressures were on children's services.
The association's study found that Welsh councils have overspent in the current financial year by a total of 13.7m.
Overspends were recorded in 15 out of the 21 authorities, with one council having overspent by 2.8m. Only two of the councils underspent.
"The primary characteristic of children's services is that it is low volume and high cost," the report said. "Therefore a small number of children who require specialist services can have a major effect on the budget."
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