A linear relationship between the shear viscosity and the total pressure, a constant single relaxation time for a Maxwell-type viscoelastic fluid, and a unidirectional velocity profile are the major assumptions made in the present work in order to study the steady-state isothermal and pressure-driven flows in straight channels and circular tubes. Despite their non-linearity the final partial differential equations that govern the flows are solved analytically, and the dependence of all the primary flow variables on the geometrical aspect ratio, the dimensionless pressure-viscosity coefficient and the Weissenberg number is revealed explicitly. It is demonstrated that the pressure-dependent viscosity slightly affects the velocity profile, changes substantially the pressure gradient along the main flow direction, generates another normal to the wall, and it is responsible for significant variations of the extra-stresses along both spatial directions. An exponential increase of the viscosity, relative to its reference value, is predicted as the distance from the exit of the channel/tube increases. As a consequence, the average pressure difference, required to drive the flow and the shear stress at the wall increase substantially compared to that predicted by the classic Hagen-Poiseuille law. Last, it is revealed that the solution of the governing equations ceases to exist when the Weissenberg number reaches a threshold. (C) 2015 Elsevier B.V. All rights reserved.