  We study new Willmore-type variational problem for a hypersurface $M$ in
$\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its
general version is the Reilly-type functional $WF_{n,s}=\int_M F(\sigma^{\cal
F}_1,\ldots,\sigma^{\cal F}_s)\,{\rm d}V$, where $\sigma^{\cal F}_i$ are
elementary symmetric functions of the eigenvalues of the second fundamental
form restricted on the leaves of $\cal F$. The first and second variations of
such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is
also shown. The Euler-Lagrange equation for a critical hypersurface with a
transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and
examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of
revolution are found, and it is shown that they are a local minimum for special
variations.
