  Time-reversal invariant topological superconductors are a new state of matter
which have a bulk superconducting gap and robust Majorana fermion surface
states. These have not yet been realized in solid state systems. In this paper,
we propose that this state can be realized in doped Weyl semimetals or Weyl
metals. The Fermi surfaces of a Weyl metal carry Chern numbers, which is a
required ingredient for such a topological superconductor. By applying the
fluctuation-exchange approach to a generic model of time-reversal invariant
Dirac and Weyl semimetals, we investigate what microscopic interactions can
supply the other ingredient, viz., sign changing of the superconducting gap
function between Fermi surfaces with opposite Chern numbers. We find that if
the normal state is inversion symmetric, onsite repulsive and exchange
interactions induce various nodal phases as well as a small region of
topological superconductivity on the phase diagram. Unlike the He$^{3}$B
topological superconductor, the phase here does not rely on any special
momentum dependence of the pairing amplitude. Breaking inversion symmetry
precludes some of the nodal phases and the topological superconductor becomes
much more prominent, especially at large ferromagnetic interaction. Our
approach can be extended to generic Dirac or Weyl metals.
