In this analysis, we apply the methods of the theory of linear absolute and convective instabilities to studying the destabilization of transverse modes in a model of convection in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow. In this first part of the analysis, normal modes are treated and neutral curves are obtained for a variety of values of the horizontal Rayleigh number, R (h), and the P,clet number, Q (v). The computations are performed by using a high-precision pseudo-spectral Chebyshev-collocation method. Our results compare well with the results found in the literature for the critical values of the vertical Rayleigh number. It is shown that the horizontal temperature gradient effect, inducing a Hadley circulation, is stabilizing for any fixed value of the throughflow velocity. The throughflow effect is stabilizing, for each of the values of R (h) = 0, 10, 20, 30. For higher values of R (h) = 40, 50, 60 considered, the influence of increasing throughflow on the stability is mixed. For a vanishing horizontal temperature gradient the critical normal mode is non-oscillatory, for all the values of throughflow. In all the cases of a non-zero horizontal temperature gradient and a non-zero throughflow considered, the critical normal mode is oscillatory, and the oscillatory frequency is an increasing function of both R (h) and Q (v).