Cold gas streaming along the dark matter filaments of the cosmic web is predicted to be the major source of fuel for disc buildup, violent disc instability and star formation in massive galaxies at high redshift. We investigate to what extent such cold gas is detectable in the extended circumgalactic environment of galaxies via Lyα absorption and selected low-ionization metal absorption lines. We model the expected absorption signatures using high-resolution zoom-in adaptive mesh refinement cosmological simulations. In the post-processing, we distinguish between self-shielded gas and unshielded gas. In the self-shielded gas, which is optically thick to Lyman continuum radiation, we assume pure collisional ionization for species with an ionization potential greater than 13.6 eV. In the optically-thin, unshielded gas, these species are also photoionized by the metagalactic radiation. In addition to absorption of radiation from background quasars, we compute the absorption line profiles of radiation emitted by the galaxy at the centre of the same halo. We predict the strength of the absorption signal for individual galaxies without stacking. We find that the Lyα absorption profiles produced by the streams are consistent with observations of absorption and emission Lyα profiles in high-redshift galaxies. Due to the low metallicities in the streams, and their low covering factors, the metal absorption features are weak and difficult to detect.