Current research indicates that the radiation of the first stars alone would not have been suficient to ionize the intergalactic medium on long distances because of the high columnar density of the clouds in which they formed. It becomes necessary, then, to explore other alternatives for the reionization. The study of the contribution of the first accreting binary systems to the reionization and heating of intergalactic medium requires the formulation of a concrete model for the first microquasars. We aim at constructing a consistent model for jets of microquasars where the donor star is of Population III. We calculate the spectral energy distribution of the radiation produced by the relativistic particles in the jets, within the framework of a lepto-hadronic model. In addition, we study the interaction between the jets and the early intergalactic medium. All calculations have been computed for a cosmological redshift value $z = 10$. We determine that jets of microquasars of Population III are sources of intense gamma radiation in regions of particle acceleration near the compact object and in the terminal jets.