The concept of magnetic nanoprobes (or magnetic nanoantennas), their excitation, and the capability of providing a magnetic near-field enhancement and vanishing electric field is presented and investigated. It is established that a particular type of cylindrical vector beam called an azimuthally electric polarized vector beam yields a strong longitudinal magnetic field on the beam axis where the electric field is ideally null. These beams, with an electric polarization vortex and cylindrical symmetry, are important in generating high magnetic to electric field contrast, i.e., large local field admittance, and in allowing selective excitation of magnetic transitions in matter located on the beam axis. We demonstrate that azimuthally polarized vector beam excitation of a photoinduced magnetic nanoprobe made of a magnetically polarizable nanocluster leads to an enhanced magnetic near field with resolution beyond the diffraction limit. We introduce two figures of merit, magnetic field enhancement and local field admittance normalized to that of a plane wave, that are useful to classify magnetic nanoprobes and their excitation. The performance of magnetic nanoprobes and azimuthal polarized beams is quantified in comparison to other illumination schemes and with several defect scenarios. The proposed probes may be useful in spectroscopy and scanning probe microscopy applications.