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The Hungaria Family (the closest region of the Main Belt to Mars) is an important source of Planet-Crossing-Asteroids and even impactors of terrestrial planets. We present the possibility that asteroids coming from the Hungaria Family get captured into co-orbital motion with the terrestrial planets in the inner solar system. Therefore we carried out long term numerical integrations (up to 100 Myr) to analyze the migrations from their original location - the Hungaria family region- into the inner solar system. During the integration time we observed whether or not the Hungarias get captured into a co-orbital motion with the terrestrial planets. Our results show that 5.5% of 200 Hungarias, selected as a sample of the whole group, escape from the Hungaria region and the probability from that to become co-orbital objects (Trojans, satellites or horseshoes) turns out to be about 3.3%: 1.8% for Mars and 1.5% for the Earth. In addition, we distinguished in which classes of co-orbital motion the asteroids get captured and for how long they stay there in stable motion. Most of the escaped Hungarias become Quasi-satellites and the ones captured as Trojans favor the L_5 lagrangian point. This work highlights that the Hungaria region is a source of Mars and also Earth co-orbital objects.