For the first time in the world practice, we have proposed transmitter-receiver systems with toroidal coils for evaluating the electrical resistivity of a cylindrical sample during core drilling. Two locations of the toroidal coils are considered: either in an insulating fiberglass pipe or in a highly conductive non-magnetic metal pipe, both inside a ferromagnetic steel outer core barrel. We have elaborated an algorithm for two-dimensional finite-difference modeling of electrical and magnetic signals from an external circular magnetic harmonic current equivalent to a toroidal coil, with regard to the magnetic permeability of the outer core barrel. Two-dimensional numerical modeling of the real (in-phase) and imaginary (quadrature) parts of the electric field vertical component and the magnetic field tangential component is conducted for typical terrigenous core resistivities. Subsequent to the numerical modeling results, the optimal lengths of the logging-while-coring systems, as well as the operating frequencies and measured signal types have been chosen when the coils are located either in the fiberglass or in non-magnetic metal pipe. Further, we come up with transforms of the measured signals into apparent resistivity values for vertically-variable core samples. Finally, we have worked out criteria for the consistency between the signals measured in thin-layered and equivalent electrically macroanisotropic cores while varying the resistivity contrast and interlayer thickness.