  In the weak-magnetic-field approximation, we derived a general expression of
hyperfine-induced Land\'{e} $g$-factors. By using this formula and the
multi-configuration Dirac-Hartree-Fock theory, the $g$-factors were calculated
for the $3s3p~^3\!P^o_0$ clock state in $^{27}$Al$^+$ and $5s5p~^3\!P^o_0$ in
$^{87}$Sr. The present results, $\delta g^{(1)}_{\rm hfs}(^3\!P^o_0) =
-1.183(6) \times 10^{-3}$ for $^{27}$Al$^+$ and $\delta g^{(1)}_{\rm
hfs}(^3\!P^o_0) = 7.78(30) \times 10^{-5}$ for $^{87}$Sr agree with
experimental values very well. Our theory is also useful to predict
hyperfine-induced Land\'e $g$-factors for other atomic systems.
