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The volume integral equations (VIEs) for electromagnetic (EM) scattering by three-dimensional (3D) penetrable objects are solved by Nystrom method. The VIEs are essential and cannot be replaced by surface integral equations (SIEs) for inhomogeneous problems, but they are usually solved by the method of moments (MoM). The Nystrom method as an alternative for the MoM has shown much promise and has been widely used to solve the SIEs, but it is less frequently applied to the VIEs, especially for 3D EM problems. In this work, we implement the Nystrom method for 3D VIEs by developing an efficient local correction scheme for singular and near singular integrals over tetrahedral elements. The scheme first interpolates the unknown functions within the tetrahedral elements and then derives analytical solutions for the resultant singular or near singular integrals after singularity subtraction. The scheme is simpler and more efficient in implementation compared with those based on the redesign of quadrature rules for the singular or near singular integrands. Numerical examples are presented to demonstrate the effectiveness of the proposed scheme and its convergence feature is also studied.