In this paper, we investigate a mathematical model of hematopoiesis, a process responsible for the regular replacement of circulating blood cells. Since differential delay equations are difficult to control analytically, numerous studies have considered the models as difference equations. The main objective of this work is to provide the necessary and sufficient conditions for oscillation. We address our problem mainly on the basis of oscillatory behavior. Moreover, the latest findings on the qualitative behavior of the biological mathematical model of discrete hematopoiesis are taken into account. More specifically, we explain the mathematical differential equation of discrete hematopoiesis. Moreover, certain significant, necessary and sufficient criteria for the solution of this discrete problem are found, which guarantee either the convergence of the non-oscillating solutions towards zero or the oscillation of all solutions of the discrete hematopoiesis model to the nonlinear lag difference with positive and negative coefficients. Some numerical examples are also given to illustrate the most important results.