The temporal linear stability analysis of shear flow of two-layer viscoelastic fluids, both described using the upper convected Maxwell (UCM) model, is carried out. The plane Couette flow under creeping flow conditions both in the absence and presence of pressure gradient is examined to study the role of unmatched elasticities in the interfacial instability. As the most critical disturbance is known to be of finite wavelength of the order of channel width, the numerical analysis examines the entire range of disturbance wavenumbers and the stability map is constructed in the parametric space of fluid Weissenberg numbers to indicate the region of stable interface. With a focus on optical fiber coating process, the analysis investigates the role of pressure gradient imposed on the plate-driven flow. Both the adverse and favorable pressure gradients are analyzed to identify the region for stable coating flow. In the presence of pressure gradient, the region of stable interface broadens when the high elasticity fluid occupies the region of low shear rate, which is the bottom (top) layer for the adverse (favorable) pressure gradient. For the two fluids of unmatched rheology, in addition to elastic instability, the role of viscosity stratification leading to a jump in the shear rate at interface is also examined. The adverse pressure gradient has a stabilizing effect when more viscous fluid is more elastic, whereas the favorable pressure gradient tends to stabilize the interface when the less viscous fluid has higher elasticity than the more viscous fluid. (C) 2015 Elsevier B.V. All rights reserved.