Cells sense and respond to mechanical forces through mechanotransduction, which regulates processes in health and disease. In single adhesive cells, mechanotransduction involves the transmission of force from the extracellular matrix to the cell nucleus, where it affects nucleocytoplasmic transport (NCT) and the subsequent nuclear localization of transcriptional regulators such as YAP. However, if and how NCT is mechanosensitive in multicellular systems is unclear. Here, we characterize and use a fluorescent sensor of nucleocytoplasmic transport (Sencyt) and demonstrate that nucleocytoplasmic transport responds to mechanics but not cell density in cell monolayers. Using monolayers of both epithelial and mesenchymal phenotype, we show that NCT is altered in response both to osmotic shocks, and to the inhibition of cell contractility. Further, NCT correlates with the degree of nuclear deformation measured through nuclear solidity, a shape parameter related to nuclear envelope tension. In contrast, YAP but NCT is sensitive to cell density, showing that YAP response to cell‐cell contacts is not via a mere mechanical effect of NCT. Our results demonstrate the generality of the mechanical regulation of NCT.