An inverse boundary element method based on the Burton-Miller integral equation is proposed for reconstructing the Neumann boundary data from pressure values on a conformal surface in the near-field of an arbitrary radiating object. The accuracy of the reconstruction is compared with that of a method based on the more commonly used Helmholtz integral equation. In particular, the behavior at characteristic frequencies, which are known to be problematic in the Helmholtz integral equation for the forward problem, is examined. The effect of regularization is considered, including the L-curve parameter selection method. Numerical computations are given for noisy data generated from an internal point source.