In this paper the spin-wave normal modes in a nanopillar with circular cross-section, excited by the simultaneous action of an external magnetic field and a spin-polarized current, are investigated. The study is performed by using a micromagnetic approach based upon a linearized Lagrangian formalism which requires to solve a complex generalized nonHermitian eigenvalue problem. In all the studied configurations it is shown that, in the linear regime and independently of the orientation of the external bias field, a spatially uniform spin-polarized current excites the lowest-frequency mode, which is not necessarily the quasi-uniform mode of the spectrum. The obtained numerical results are also confirmed by finite-difference time-domain numerical simulations.