We generalize pressure boundary conditions of an ε-Stokes problem. Our ε-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter ε>0. For the Dirichlet boundary condition, it is proven in Matsui and Muntean (Adv Math Sci Appl, 27:181–191,2018) that the solution for the ε-Stokes problem converges to the one for the Stokes problem as ε tends to 0, and to the one for the pressure-Poisson problem as εtends to ∞. Here, we extend these results to the Neumann and mixed boundary conditions. We also establis herror estimates in suitable norms between the solutions to the ε-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in ε. Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the ε-Stokes problem has a nice asymptotic structure.