Lateral Josephson junctions (LJJ) made of two superconducting Nb electrodes coupled by Cu-film are applied to quantify the stray magnetic field of Co-coated cantilevers used in magnetic force microscopy (MFM). The interaction of the magnetic cantilever with LJJ is reflected in the electronic response of LJJ as well as in the phase shift of cantilever oscillations, simultaneously measured. The phenomenon is theorized and used to establish the spatial map of the stray field. Based on our findings, we suggest integrating LJJs directly on the tips of cantilevers and using them as nanosensors of local magnetic fields in scanning probe microscopes. Such probes are less invasive than conventional magnetic MFM cantilevers and simpler to realize than SQUID-on-tip sensors.