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The field of tissue engineering has made significant advancements with extrusion-based bioprinting, which uses shear forces to create intricate tissue structures. However, the success of this method heavily relies on the rheological properties of bioinks. Most bioinks use shear-thinning. While a few component-based efforts have been reported to predict the viscosity of bioinks, the impact of shear rate has been vastly ignored. To address this gap, our research presents predictive models using machine learning (ML) algorithms, including polynomial fit (PF), decision tree (DT), and random forest (RF), to estimate bioink viscosity based on component weights and shear rate. We utilized novel bioinks composed of varying percentages of alginate (2–5.25%), gelatin (2–5.25%), and TEMPO-Nano fibrillated cellulose (0.5–1%) at shear rates from 0.1 to 100 s−1. Our study analyzed 169 rheological measurements using 80% training and 20% validation data. The results, based on the coefficient of determination (R2) and mean absolute error (MAE), showed that the RF algorithm-based model performed best: [(R2, MAE) RF = (0.99, 0.09), (R2, MAE) PF = (0.95, 0.28), (R2, MAE) DT = (0.98, 0.13)]. These predictive models serve as valuable tools for bioink formulation optimization, allowing researchers to determine effective viscosities without extensive experimental trials to accelerate tissue engineering.