Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of [Formula: see text]-rational fractal interpolation surfaces (FISs) on a rectangular grid in the present paper that contain scaling factors in both directions and two types of positive real parameters which are referred as shape parameters. The graphs of these [Formula: see text]-rational FISs are the attractors of suitable rational iterated function systems (IFSs) in [Formula: see text] which use a collection of rational IFSs in the [Formula: see text]-direction and [Formula: see text]-direction and hence these FISs are self-referential in nature. Using upper bounds of the interpolation error of the [Formula: see text]-direction and [Formula: see text]-direction fractal interpolants along the grid lines, we study the convergence results of [Formula: see text]-rational FISs toward the original function. A numerical illustration is provided to explain the visual quality of our rational FISs. An extra feature of these fractal surface schemes is that it allows subsequent interactive alteration of the shape of the surfaces by changing the scaling factors and shape parameters.