Tissue engineering aims at recreating tissues that can be employed to replace lost or damaged tissues, or at producing in laboratory meat destined for food use without the need for slaughtering animals. Tissue growth is more favorable if cells are encapsulated within hydrogel scaffolds, which provide them a mechanically supportive environment that mimics the extracellular matrix in which they are immersed in vivo. Among others, a key factor for a successful tissue growth is cell motility, which in turn depends on many aspects, such as spatiotemporal changes of cellular microenvironment due to scaffold degradation, chemical stimuli induced by other neighbouring cells triggering specific cell-cell signalling pathways, proper transportation of growth factors, oxygen, nutrients, and waste in/out of the constructs. To maximize the quality and quantity of the final product, it is necessary to optimize the process by finely tuning all the relevant variables. However, such optimization is unfeasible through trial-and-error approaches, since the nature of the tests would lead to long lead times and prohibitive costs. Numerical models of cell motility and tissue growth can be used as a relatively fast and low-cost alternative to successfully and effectively simulate in-vitro experiments. With this background, in the present work a novel computational model is developed to simulate cell motion in bioprinted scaffolds up to the formation of a cluster, considering precursor muscle cells (myoblasts) and PEG-fibrinogen-based hydrogels as a case study. Cell migration is treated as an advective/diffusive process modelled via the phase-field approach. Governing equations of the phase-field approach are coupled with transport equations introduced to model various chemo-biological mechanisms involved in the process, such as scaffold degradation, cellular metabolic functions and chemoattractant and nutrient diffusion through the hydrogel. Numerical results highlight the role of the nutrient and chemoattractant availability in the construct, as well as of the interplay between cell motion and scaffold chemo-mechanical properties. The developed computational framework represents then a first step towards a predictive tool for tissue growth in bioprinted scaffolds, and it is eligible to be exploited in the context of the optimization of culture medium in which bioprinted constructs are immersed during the tissue growth phase.