The rutile germanium oxide semiconductor (rutile-GeO2) could act as a half-metallic compound and be promising for spintronic and optoelectronic applications. The calculations have been performed within Korringa-Kohn-Rostoker (KKR) approach combined with the coherent potential approximation (CPA) which are combined with two approximations, the Local Density-Approximation (LDA) then the self-interaction corrected LDA approximation (LDA-SIC) in order to study the electronic structure of bulk rutile-GeO2 doped and co-doped with three transition metals impurities Fe, Co and Ni. The doping value is set to a value of 10% while the co-doping is set to 5% for each impurity. The main findings in this work are: (1) A direct ultrawide bandgap of 4.80 eV is observed and the rutile-GeO2 exhibits N-type semiconducting property. (2) The doped and co-doped GeO2 acquires a magnetic behavior and shows a half-metallicity feature. (3) The mechanism responsible of this property is also studied. (4) The critical temperature can achieve 334 K when doped with Fe while may raises to 398 K when it is co-doped with Fe and Co. (5) The spin polarization can be improved with co-doping. It can be inferred that rutile-GeO2 doped or co-doped with (Co, Fe) transition metals can be considered as a potential candidate for spintronic and optoelectronic applications.