We investigate the role of thermal conductivity (k) of dispersed nanomaterial on the thermal and rheological properties of metal and metal oxide nanofluids with average particles size (similar to 7 nm), stabilized with a monolayer of surfactant. Iron oxide and silver nanoparticles have been synthesized by coprecipitation and reduction techniques, respectively. Our studies confirm that the thermal conductivity of nanoparticle does not influence the k enhancement in stable nanofluids at low particle loading. In such nanofluids, the k enhancement follows the effective medium theory (i.e., lower Hashin and Shtrikrnan bounds) where both the k and viscosity follow a time independent behavior. Our results confirm that in the dilute limit, the k of nanofluids is solely dependent on the volume fraction of nanoparticle. The present results also suggest that the surface functionalization of nanoparticles is the key to obtain time independent thermophysical properties, which is important for practical applications.