We predict that unpolarized charge current injected into a ballistic thin film of prototypical topological insulator (TI) will generate a noncollinear spin texture on its surface. Furthermore, the nonequilibrium spin texture will extend into an -nm-thick layer below the TI surfaces due to penetration of evanescent wave functions from the metallic surfaces into the bulk of TI. Averaging over a few angstroms along the longitudinal direction defined by the current flow reveals a large component pointing in the transverse direction. In addition, we find an order of magnitude smaller out-of-plane component when the direction of injected current with respect to Bi and Se atoms probes the largest hexagonal warping of the Dirac-cone dispersion on the TI surface. Our analysis is based on an extension of the nonequilibrium Green's functions combined with density functional theory (NEGF+DFT) to situations involving noncollinear spins and spin-orbit coupling. We also demonstrate how DFT calculations with a properly optimized local orbital basis set can precisely match putatively more accurate calculations with a plane-wave basis set for the supercell of .