The influence of partial replacement of Fe with Co in the quaternary (Fe71.2B24Y4.8)96Nb4 bulk metallic glasses on their structure, thermal stability and magnetic properties was studied. It was found that Co increases the thermal stability, as well as the Curie temperature, which monotonously increases as the Co content increases. The saturation magnetization shows a maximum of 1.01 μB per magnetic atom for x = 0.1, followed by a perfectly linear decrease for higher Co contents. The extension of the supercooled liquid region may reach even 98 K and the glass transition temperatures approach the theoretical value of 2/3 of the melting temperature. The mean field theory allows to calculate the magnetic exchange stiffness constant and to correlate its variation with the variation of the magnetic saturation.