In situ stress and medium-energy electron-diffraction (MEED) measurements have been performed simultaneously during the deposition of FeMn on Cu(001). For a thickness above 5 layers, stress and MEED exhibit coherent layer-by-layer oscillations with a period of one atomic layer, where the largest compressive stress corresponds to the filled layer. In this thickness regime, the average stress is -0.59±0.02 GPa. From this, we deduce the biaxial modulus of FeMn layers as 148 (±5) GPa, which agrees well with the respective bulk value. For a FeMn thickness below 1.5 layers, the resulting stress is qualitatively ascribed to the sum of the individual stress contributions from Fe on Cu(001) and Mn on Cu(001). A c(2x2) low-energy electron diffraction pattern in this thickness regime indicates the formation of a c(2x2) MnCu surface alloy in the initial growth of FeMn on Cu(001), which induces a compressive surface stress of -0.7 N/m for the initial deposition of the FeMn alloy. This surface alloy formation leads to a Fe-rich FeMn alloy near the Cu interface. This compositional change might modify the antiferromagnetic coupling of the 1:1 FeMn alloy.