For highly viscous fluids that slip in parallel sliding plate rheometers, we want to use a slightly converging flow to suppress this wall slip. In this work, we first attack the steady shear flow of a highly viscous Newtonian fluid between two gently converging plates with no slip boundaries using the equation of motion in cylindrical coordinates, which yields no analytical solution. Then we treat the same problem using the lubrication approximation in Cartesian coordinates to yield exact, explicit solutions for dimensionless velocity, pressure and shear stress. This work deepens our understanding of a drag flow through a gently converging slit of arbitrary convergence angle. We also employ the corotational Maxwell model to explore the role of viscoelasticity in this converging shear flow. We then compare these analytical solutions to finite element calculations for both Newtonian and corotational Maxwell cases. A worked example for determining the Newtonian viscosity using a converging shear rheometer is also included. With this work, we provide the framework for exploring other constitutive equations or other boundary conditions in future work. Our results can also be used to design the linear bearings used for the parallel sliding plate rheometer (SPR). This work can also be used to evaluate the error in the shear stress that is caused by bearing misalignment and specify the parallelism tolerance for the linear bearings incorporated into a SPR.