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 A0 <= 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.