The usually assumed identification of small-scale magnetic concentrations with bright facular or network points on the photosphere is observationally checked by using high spatial resolution spectra of Ca II K bright points very near the disk center. The detection of spatially unresolved magnetic structures is made via a new differential analysis of the well-known pair of Fe I lines 5247.06 A and 5250.22 A; these concentrations are present in the central part of a line weakening zone, which is of some 2 arcsec wide. No continuum intensity enhancement with respect to the quiet photosphere can be ascertained of these structures, within an accuracy threshold of 1.2 percent. In spite of this, magnetic concentrations brighter than the quiet photosphere are compatible with the observations, but if so, they must be narrower than 0.2 arcsec.