  The $\Xi$ single-particle potential obtained in nuclear matter with the
next-to-leading order baryon-baryon interactions in chiral effective field
theory is applied to finite nuclei by an improved local-density approximation
method. As a premise, phase shifts of $\Xi N$ elastic scattering and the
results of Faddeev calculations for the $\Xi NN$ bound state problem are
presented to show the properties of the $\Xi N$ interactions in the present
parametrization. First, the $\Xi$ states in $^{14}$N are revisited because of
the recent experimental progress, including the discussion on the $\Xi N$
spin-orbit interaction that is relevant to the location of the $p$-state. Then
the $\Xi$ levels in $^{56}$Fe are calculated. In particular, the level shift
which is expected to be measured experimentally in the near future is
predicted. The smallness of the imaginary part of the $\Xi$ single-particle
potential is explicitly demonstrated.
