Several flows of practical interest are of viscoelastic fluids and it is desirable to know if these flows are in a laminar or turbulent state. In this paper, the laminar-turbulent transition is studied in which the convection of Tollmien-Schlichting waves is investigated for the incompressible two-dimensional flow between two parallel plates. The viscoelastic fluid adopted is modeled by the Oldroyd-B constitutive equation. Direct Numerical Simulation (DNS) and Linear Stability Theory (LST) were used to verify the stability of the viscoelastic fluid flow to unsteady disturbances. In the DNS formulation, the governing equations are discretized by high-order compact finite difference schemes for the spatial derivatives, while time integrations are carried out by a fourth order Runge Kutta method. For the LST analysis, the Orr-Sommerfeld equation is modified to include viscoelastic effects and solved by a shooting method. In order to evaluate the flow stability when submitted to unsteady disturbances and to find the neutral stability curves, a range of numerical simulations was performed with different dimensionless parameters for the viscoelastic fluid flow and the results were compared with the Newtonian fluid flow. The influence of the elastic forces and the amount of the polymer concentration in the fluid were investigated. A wide range of polymer concentration was studied. The results show that the variation of the critical Reynolds number may be monotonic for certain conditions. The DNS simulations were carried out, and the amplification rates obtained were compared with the LST results, with a very good agreement. The DNS results allow also the visualization of the disturbances variation in streamwise and wall-normal directions, showing the behavior of the velocity components, vorticity and the non-Newtonian extra-stress tensor components. The DNS results show a significant change in flow structure for certain combinations of the polymer concentration and elastic forces parameters.