The formation of preferential flow paths during infiltration of water into homogeneous, dry soil is an important phenomenon whose explanation and prediction have remained elusive under the standard theories of multiphase flow in porous media. We have recently proposed a macroscopic phase-field model of unsaturated flow in porous media, which explains why such fingering occurs [L. Cueto-Felgueroso and R. Juanes, Phys. Rev. Lett. 101, 244504 (2008)]. Here we present a linear stability analysis of the proposed model for constant-flux infiltration, which allows a quantitative description of the wetting front instability. The present analysis stresses the critical role of the initial water saturation and applied flux ratio in the asymptotic stability of the system, as well as in the transient growth of perturbations, which is consistent with the experimental evidence. The trends in the frequency and growth factor of the most unstable modes predicted by our analysis are also in quantitative agreement with experimental measurements.