  Recently symmetries of gravity and gauge fields in the asymptotic regions of
spacetime have been shown to play vital role in their low energy scattering
phenomena. Further, for the black hole spacetime, near horizon symmetry has
been observed to play possible role in understanding the underlying degrees of
freedom for thermodynamic behaviour of horizon. Following the similar idea, in
this paper, we analyzed the symmetry and associated algebra near a timelike
surface which is situated at any arbitrary radial position and is embedded in
black hole spacetime. In this paper we considered both Schwarzschild and Kerr
black hole spacetimes. The families of hypersurfaces with constant radial
coordinate (outside the horizon) in these spacetimes is timelike in nature and
divide the space into two distinct regions. The symmetry algebra turned out to
be reminiscent of Bondi-Metzner-Sach (BMS) symmetry, found in the asymptotic
null infinity.
