This paper is concerned with the global well-posedness of classical solutions to a system in three dimension. The system is derived from a Keller-Segel type repulsive chemotaxis model with singular sensitivity, but the chemical production rate is nonlinear. We establish the existence global in time and uniqueness of classical solutions to the Cauchy problem for this chemotaxis model by splitting the equations in a linear and a nonlinear part and introducing an improved energy method. To this end, we introduce the monotonicity estimates because the supercritical case is involved.