The possible existence of noticeable fluxes of antiparticles in the Earth magnetosphere has been predicted on theoretical considerations in this article. The antiprotons expected at several hundred kilometers of altitudes, we do not believe are of direct extraterrestrial origin, but are the natural products of nuclear reactions of the high-energy primary cosmic rays (CR) with the constituents of the terrestrial atmosphere. Extraterrestrial, galactic antiprotons are themselves of secondary in origin, i.e. they are born in nuclear reactions of the same CR particles passing through 5--7 g/cm2 of interstellar matter encountered during their lifetime in the Galaxy. We expect that the fluxes of magnetospheric antiprotons to be higher compared to the interstellar fluxes because the fluxes get accumulated due to confinement by the magnetic field of the Earth. We present the results of the computations of the antiproton fluxes at 50 MeV to several GeV energies due to the CR particle interactions with the matter in the interstellar space, and also with the residual atmosphere at altitudes of ~1000 km over the Earth's surface. The estimates show that the magnetospheric antiproton fluxes are two orders of magnitude greater compared to the interstellar fluxes measured at energies < 1 GeV.