Three-dimensional (3D) eddy current non-destructive evaluation of steam generator tubes is cast in a wavefield integral framework and considered from an inverse problem point of view. Three inversion algorithms based on a gradient-type iterative search are studied: the modified gradient method, an inversion via the source-type integral equation, and one based on the localized nonlinear approximation. To overcome the inherent ill-posed nature of the inversion problem, a binary constraint is enforced on the contrast to be retrieved. Special attention is devoted to the evolution of the iterative processes, which is controlled by means of a tunable 'cooling' parameter. The difficulty of such 3D reconstructions is discussed with emphasis on attenuation problems that are typical in eddy current analysis.