This paper is devoted to obtaining and analysis of steady-state solutions of one-dimensional equations for the simulation of blood flow when the non-Newtonian nature of blood is taken into account. The models, based on the rheological relations, widely used for the blood, are considered. The expressions for the nonlinear frictional term are presented. For the Power Law, Simplified Cross, and Quemada models, the exact integrals of the nonlinear ordinary differential equation, obtained from the averaged momentum equation, are obtained. It is demonstrated that several solutions exist for every rheological model, but the physically relevant solutions can be selected by the appropriate value of Mach number. The effects of the velocity profile and the value of hematocrit on the steady-state solutions are analyzed. It is demonstrated that the flattening of the velocity profile, which is typical for the blood, leads to the diminishing of the length of the interval, where the solution exists. The same effect is observed when the hematocrit value is increased.