The stability, electronic and magnetic properties of Fe atoms adsorbed on the self-assembled Bi-dimer lines nanostructure on the H/Si(001) surface are addressed by spin-density functional calculations. Our results show that Fe adatoms are much more stable on sites closer to the Bi nanolines being able to form one-dimensional atomic arrays. The most stable structure occurs on a missing dimer line aside the Bi dimers, which corresponds to an array with distances between Fe adatoms of about 8 Ang. In this array the irons are coupled antiferromagnetically with spin magnetic moment of about 1.5 Bohr magnetons per Fe atom, whereas the coupling exchange interactions is found to be of 14.4 meV. We also estimate a large magnetic anisotropy energy for the Fe adatom of about 3 meV/atom. In addition, the electronic band structure of the Fe array at the most stable structure shows a magnetic half-metal behavior.