Bioprinting is a rapidly advancing technology that allows the recreation of functional tissues in the laboratories starting from stem cells, and has the potential to revolutionize many application fields, like medicine and the food sector. Tissue growth is more easily achievable if cells are embedded in hydrogel scaffolds, which provide them a mechanically supportive environment and ensure their viability. The neo-tissue growth in hydrogels is a complex process determined by a number of factors, among which effective cellular clustering is a crucial one. As such, the efficient development of new tissue is strongly influenced by cell motion. This in turn depends on several factors, like chemical and/or mechanical stimuli triggered by cellular micro-environment, spatio-temporal variations in the hydrogel properties induced by enzymes secreted by cells, and proper transport of growth factors, oxygen, nutrients and waste in/out of the constructs. To maximize the final mass, so as to make the process scalable, a proper adjustment of the main variables should be performed, especially in the post-printing phase due to the high costs of cellular culture medium. However, such optimization is hardly feasible through trial-and-error approaches, as the indeterminate nature of the test results in long lead times and prohibitive costs. Hence, numerical models of cell motility can be used as a quick and valid alternative to successfully and effectively simulate in vitro experiments. In this context, we present a new computational model for cell motion in enzyme-sensitive bioprinted hydrogel scaffolds, up to the compaction in clusters. Cell motility is treated as a moving boundary problem, tackled via a phase-field (PF) approach. The diffusive/advective PF equations are coupled with other key phenomena characteristic of the process, such as: i) Nutrient diffusion through the construct and its metabolism by cells; ii) Expression of chemoattractant substances triggering chemotaxis, and of enzymes inducing local scaffold degradation; iii) Environment mechano-sensing processes, which drive cell motion along positive stiffness paths (durotaxis). Obtained results showcase the pivotal importance of the cell micro-environment properties (in terms of stiffness and availability of the relevant chemical species) for cell crawling in hydrogel scaffolds. The proposed computational framework thus paves the way towards the development of a predictive optimization tool.