Invisibility dips, i.e., minima in scattering spectrum associated with asymmetric Fano-like line-shapes, have been predicted with transformation optics in studying strong coupling between two plasmonic nanoparticles. This feature of strongly coupled plasmonic nanoparticles holds promise for sensor cloaking. It requires an extremely narrow gap between the two nanoparticles, preventing its experimental observation at optical frequencies. Here, the concept of spoof surface plasmons is used to facilitate the strong coupling between two spoof-localized-surface-plasmon (SLSP) resonators. Instead of observing in far field, the near-field energy transport is probed through the two SLSP resonators. By virtue of enhanced coupling between the two resonators stacked vertically, a spectral transmission dip with asymmetric Fano-like line-shape, similar to the far-field "invisibility dips" predicted by transformation optics, is observed. The underlying mode interference mechanism is further demonstrated by directly imaging the field maps of interfered waves that are tightly localized around the resonators. These near-field "invisibility dips" may find use in near-field sensing, on-chip switching, filters, and logical operation elements.