The ultimate conductometric sensor for ferromagnetic activity of nanoscale magnetic materials could be a single carbon nanotube. We show that the electrical conductance of an individual carbon nanotube is sensitive to magnetic transitions of nanoscale magnets embedded inside it. To establish this, multiwall carbon nanotubes were impregnated with cobalt nanoclusters. Temperature dependence of conductance (5 K < T <300 K) of these nanotubes shows the usual Lüttinger-liquid power law behavior at higher temperatures and an onset of Coulomb blockade at lower temperatures. At the lowest temperature (T approximately 6 K), the differential conductance (dI/dV versus V) develops aperiodic fluctuations under an external magnetic field B, the rms amplitude of which grows with the magnitude of the field itself. Low-temperature magnetoconductance, studied as function of temperature and bias, can be interpreted in terms of weak antilocalization effects due to the presence of the magnetized clusters. The temperature dependence of magnetoconductance further presents a "peak"-like feature and slow dynamics around T =55 K, which depend on the magnitude and history of the applied B field. These observations indicate a sensitivity of electronic transport in the multiwall nanotubes to the dynamics of nanoscale magnets at low temperature.