Sustained proliferation is a significant driver of cancer progression. Cell-cycle advancement is coupled with cell size, but it remains unclear how multiple cells interact to control their volume in 3D clusters. In this study, we propose a mechano-osmotic model to investigate the evolution of volume dynamics within multicellular systems. Volume control depends on an interplay between multiple cellular constituents, including gap junctions, mechanosensitive ion channels, energy-consuming ion pumps, and the actomyosin cortex, that coordinate to manipulate cellular osmolarity. In connected cells, we show that mechanical loading leads to the emergence of osmotic pressure gradients between cells with consequent increases in cellular ion concentrations driving swelling. We identify how gap junctions can amplify spatial variations in cell volume within multicellular spheroids and, further, describe how the process depends on proliferation-induced solid stress. Our model may provide new insight into the role of gap junctions in breast cancer progression.