An investigation of possibilities for superconductivity mediated by spin fluctuations in some elementary metals is motivated by the recent discovery of superconductivity in the hcp high-pressure phase of iron. The electronic structure, the electron-phonon coupling (lambda(ep)) and the coupling due to spin fluctuations (lambda(sf)) are calculated for different phases and different volumes for four elementary metals. The results show that such possibilities are best for systems near, but on the non-magnetic side of, a magnetic instability. Fcc Ni, which show stable magnetism over a wide pressure range, is not interesting in this respect. Ferromagnetic and anti-ferromagnetic fluctuations in hcp Fe contribute to a relatively strong coupling in the pressure range where superconductivity is observed. The absence of fluctuations at large q-vectors makes fcc Pd only moderately interesting despite its large exchange enhancement for q = 0. Fcc Co at high pressure (similar to0.5 Mbar) behaves as an improved version of Pd, where the fluctuations extend to larger q. The estimations of Tc, which reproduce the experimental situation in Fe quite well, suggest a measurable Tc for the high-pressure phase of fcc Co, while the estimate is lower for the ambient-pressure phase of fcc Pd. (C) 2002 Elsevier Science B.V. All rights reserved.