We have performed extensive molecular dynamics simulations of nanoindentation of an ice slab with model atomic force microscopy (AFM) tips. We found the presence of a quasi-liquid layer between the tip and the ice for all explored indentation depths. For the smallest tip studied (R = 0.55 nm), the force versus indentation depth curves present peaks related to the melting of distinct monolayers of ice, and we were able to calculate the work (free energy) associated with it. For a larger tip (R = 1.80 nm) having a size not commensurate with the average monolayer thickness, we did not find a clear structure in force curves. This work can help guide the interpretation of experimental AFM indentation of ice and other crystalline solids. More specifically, it provides guidelines for tip sizes where layer-by-layer melting can be achieved and for the order of magnitude of forces that need to be detected.