Oldroyd-B and FENE-CR constitutive equations are employed for isothermal viscoelastic fluids as an extension to the previously developed Arbitrary Lagrangian Eulerian (ALE) approach [2, 3]. The governing equations are discretized using a div-stable, side-centered finite volume approach in which the velocity components are defined at the mid-points of element faces, the displacement vector is defined at the vertices, and the pressure and modified conformation tensor are defined at the element centroids. The pressure and modified conformation tensor fields are treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. Surface tension force is treated as a tangent force and across the interface the jump conditions are satisfied exactly. For the computation of unit normal vector, the mean weighted by sine and edge length reciprocals (MWSELR) approach is implemented because the parasitic currents are found to be very sensitive to the computation of normal vectors [3]. The mass conservation of both species has been achieved at machine precision by giving a special attention to enforce the kinematic boundary condition at the interface in the normal direction, while obeying the discrete geometric conservation law (DGCL). The resulting algebraic equations are solved in a fully coupled (monolithic) manner and an approximate matrix factorization similar to that of the projection method is employed. The parallel algebraic multigrid solver BoomerAMG provided by the HYPRE library (accessed through the PETSc library) has been utilized for the scaled discrete Laplacian of pressure and the diagonal blocks of mesh deformation equations. The classical benchmark problem of a buoyancy-driven motion of a single rising Newtonian bubble in a viscoelastic fluid is presented [1, 4, 5, 6]. Behind the bubble, positive and negative transient wakes are observed which shows that the shear-thinning behavior of a viscoelastic fluid model is not essential for the formation of a transient negative wake.