Topological channels are typically pinned to physical edges or heterogeneous interfaces. Physical edges expose the channels to localized disorder, whereas interface-based designs generally involve material or structural discontinuities. These constraints limit geometric tunability and can compromise microscopic transport quality. Here we show that smooth strain can program topological channels within a single-component homogeneous lattice through a mechanically constrained inverse-design framework. A strain-dependent Dirac mass provides a direct geometric control principle: its zero contour defines the channel path, the contour-crossing topological mismatch fixes the net chirality, and the normal mass gradient sets the confinement width following an inverse-square-root scaling law. Unlike conventional pinned channels, these strain-engineered interior channels permit continuous control over channel geometry and confinement. We demonstrate the approach in a strained Haldane model, realizing straight channels at prescribed orientations, designed curved paths, and multichannel networks associated with higher-Chern-number phases. The resulting channels exhibit quantized transmission and spatial decoupling from boundary-localized disorder, establishing strain as a mechanical design field for programmable interior topological transport.