Monolayer black phosphorus (MBP) is an interesting emerging electronic material with a direct band gap and relatively high carrier mobility. In this work we report a theoretical investigation of nonequilibrium spin injection and spin-polarized quantum transport in MBP from ferromagnetic Ni contacts, in two-dimensional magnetic tunneling structures. We investigate physical properties such as the spin injection efficiency, the tunnel magnetoresistance ratio, spin-polarized currents, charge currents and transmission coefficients as a function of external bias voltage, for two different device contact structures where MBP is contacted by Ni(111) and by Ni(100). While both structures are predicted to give respectable spin-polarized quantum transport, the Ni(100)/MBP/Ni(100) trilayer has the superior properties where the spin injection and magnetoresistance ratio maintains almost a constant value against the bias voltage. The nonequilibrium quantum transport phenomenon is understood by analyzing the transmission spectrum at nonequilibrium.