  We study behaviour of gravitational waves in the recently introduced general
relativistic polytropic spheres containing a region of trapped null geodesics
extended around radius of the stable null circular geodesic that can exist for
the polytropic index $N>2.138$ and the relativistic parameter, giving ratio of
the central pressure $p_\mathrm{c}$ to the central energy density
$\rho_\mathrm{c}$, higher than $\sigma = 0.677$. In the trapping zones of such
polytropes, the effective potential of the axial gravitational wave
perturbations resembles those related to the ultracompact uniform density
objects, giving thus similar long-lived axial gravitational modes. These
long-lived linear perturbations are related to the stable circular null
geodesic and due to additional non-linear phenomena could lead to conversion of
the trapping zone to a black hole. We give in the eikonal limit examples of the
long-lived gravitational modes, their oscillatory frequencies and slow damping
rates, for the trapping zones of the polytropes with $N \in (2.138,4)$.
However, in the trapping polytropes the long-lived damped modes exist only for
very large values of the multipole number $\ell>50$, while for smaller values
of $\ell$ the numerical calculations indicate existence of fast growing
unstable axial gravitational modes. We demonstrate that for polytropes with $N
\geq 3.78$, the trapping region is by many orders smaller than extension of the
polytrope, and the mass contained in the trapping zone is about $10^{-3}$ of
the total mass of the polytrope. Therefore, the gravitational instability of
such trapping zones could serve as a model explaining creation of central
supermassive black holes in galactic halos or galaxy clusters.
