  In this paper we consider the non-Abelian T-dual geometry of the type $IIB$
supergravity theory on $AdS_3\times S^3\times T^4$ background along a
convenient $SU(2)$ subgroup of the $SO(4)$ R-symmetry. We examine various null
geodesics of the resulting massive type $IIA$ supergravity theory and
investigate the Penrose limits along these geodesics. We find that one of the
resulting backgrounds admits pp-wave geometry in the neighbourhood of a
suitable null geodesic. We carry out the supersymmetry analysis of the
resulting pp-wave geometry and observe that it preserves sixteen supercharges.
Further we comment on the possible gauge theory dual of the resulting pp-wave
background.
