  We investigate interacting spin susceptibilities in lattice models for
$\mathcal{T}$-reversal symmetry-broken Weyl semimetals. We employ a random
phase approximation (RPA) method for the spin-SU(2)-symmetry-broken case that
includes mixtures of ladder and bubble diagrams, beyond a SU(2)-symmetric case.
Within this approach, the relations between the tendency towards magnetic order
and the band structure tilt parameter $\gamma$ under different temperatures are
explored. The critical interaction strength $U_c$ for magnetic ordering
decreases as the tilt term changes from type-I Weyl semimetals to type-II. The
lower temperature, the sharper is the drop in $U_c$ at the critical point
between them. The variation of $U_c$ with a slight doping near half-filling is
also studied. It is generally found that these Weyl systems show a strongly
anisotropic spin response with an enhanced doubly degenerate transverse
susceptibility perpendicular to tilt direction, inherited from
$\mathcal{C}_{4z}$ rational symmetry of bare Hamiltonian, but with the
longitudinal response suppressed with respect to that. For small tilts $\gamma$
and strong enough interaction, we find two degenerate ordering patterns with
spin order orthogonal to the tilt direction but much shorter spin correlation
length parallel to the spin direction. With increasing the tilt, the system
develops instabilities with respect to in-plane magnetic orders with wavevector
$(0,\pi, q_z)$ and $(\pi,0, q_z)$, with $q_z$ increasing from 0 to $\pi$ before
the transition to a type-II Weyl semimetal is reached. These results indicate a
greater richness of magnetic phases in correlated Weyl semimetals that also
pose challenges for precise theoretical descriptions.
