The pressure and temperature dependence of the yield stress is well established in oil drilling. In the present work, the steady, annular pressure-driven flow of a Bingham plastic is considered under the assumption that both the plastic viscosity and the yield stress vary linearly with pressure. A semi-analytical solution is derived for the case where the growth coefficients of both rheological parameters are equal, which, for certain oil drilling fluids, is indeed a reasonable assumption allowing the existence of a separable solution with an annular unyielded core. The inner and outer radii of the unyielded core are determined by solving numerically an algebraic system of equations. The pressure, which is two-dimensional in the yielded parts of the domain, and the velocity, which varies only with the radius, are calculated by means of explicit analytical expressions. The conditions for the occurrence of flow and the effects of the growth coefficient and the radii ratio on the width of the unyielded annular plug, the velocity profiles, and the pressure distributions are discussed.