Summary: We propose a new concept, the transversely trapping surface (TTS), as an extension of the static photon surface characterizing the strong gravity region of a static/stationary spacetime in terms of photon behavior. The TTS is defined as a static/stationary timelike surface \(S\) whose spatial section is a closed two-surface, such that arbitrary photons emitted tangentially to \(S\) from arbitrary points on \(S\) propagate on or toward the inside of \(S\). We study the properties of TTSs for static spacetimes and axisymmetric stationary spacetimes. In particular, the area \(A_0\) of a TTS is proved to be bounded as \(A_0\le 4\pi(3GM)^2\) under certain conditions, where \(G\) is the Newton constant and \(M\) is the total mass. The connection between the TTS and the loosely trapped surface proposed by us [Prog. Theor. Exp. Phys. 2017, 033E01 (2017)] is also examined.