  This is the first of series of papers in which we investigate stability of
the spherically symmetric space-time with de Sitter center. Geometry,
asymptotically Schwarzschild for large $r$ and asymptotically de Sitter as
$r\to 0$, describes a vacuum nonsingular black hole for $m\geq m_{cr}$ and
particle-like self-gravitating structure for $m < m_{cr}$ where a critical
value $m_{cr}$ depends on the scale of the symmetry restoration to de Sitter
group in the origin. In this paper we address the question of stability of a
vacuum non-singular black hole with de Sitter center to external perturbations.
We specify first two types of geometries with and without changes of topology.
Then we derive the general equations for an arbitrary density profile and show
that in the whole range of the mass parameter $m$ objects described by
geometries with de Sitter center remain stable under axial perturbations. In
the case of the polar perturbations we find criteria of stability and study in
detail the case of the density profile $\rho(r)=\rho_0 e^{-r^3/r_0^2 r_g}$
where $\rho_0$ is the density of de Sitter vacuum at the center, $r_0$ is de
Sitter radius and $r_g$ is the Schwarzschild radius.
