Energetic neutral atoms (ENAs) are recognized as a powerful tool for remote probing of distant hot space plasmas, in particular the plasmas abundant at the heliospheric boundary where the expanding solar wind meets the surrounding local interstellar medium. We show here for the first time that the heliospheric ENAs originating in the heliospheric sheath between the termination shock and the heliopause and reaching the inner heliosphere also provide an important and heretofore unaccounted source of atomic hydrogen in the Sun's vicinity. These ENAs are a major contributor to the density of interplanetary hydrogen at heliocentric distances <1 AU and could dominate in the downwind (interstellar wind) region under typical solar and interstellar conditions. Mass transport by heliospheric ENAs may become especially important for determining the origin of the pickup ions attributed to the inner source of neutral particles in the Sun's vicinity and for characterization of the three-dimensional solar wind flow by imaging in extreme ultraviolet.