The modulus of elasticity of soft materials on the nanoscale is of interest when studying thin films, nanocomposites, and biomaterials. Two novel modes of atomic force microscopy (AFM) have been introduced recently: HarmoniX and PeakForce QNM. Both modes produce distribution maps of the elastic modulus over the sample surface. Here we investigate the question of how quantitative these maps are when studying soft materials. Three different polymers with a macroscopic Young's modulus of 0.6-0.7 GPa (polyurethanes) and 2.7 GPa (polystyrene) are analyzed using these new modes. The moduli obtained are compared to the data measured with the other commonly used techniques, dynamic mechanical analyzer (DMA), regular AFM, and nanoindenter. We show that the elastic modulus is overestimated in both the HarmoniX and PeakForce QNM modes when using regular sharp probes because of excessively overstressed material in the samples. We further demonstrate that both AFM modes can work in the linear stress-strain regime when using a relatively dull indentation probe (starting from similar to 210 nm). The analysis of the elasticity models to be used shows that the JKR model should be used for the samples considered here instead of the DMT model, which is currently implemented in HarmoniX and PeakForce QNM modes. Using the JKR model and similar to 240 nm AFM probe in the PeakForce QNM mode, we demonstrate that a quantitative mapping of the elastic modulus of polymeric materials is possible. A spatial resolution of similar to 50 nm and a minimum 2 to 3 nm indentation depth are achieved.