Significant volumes of water are injected into the subsurface for purposes such as maintaining reservoir pressure, enhancing production efficiency, or water disposal. In these operations, injection pressures are typically kept low to prevent the formation from fracturing. However, fractures may still be induced even at low injection pressures if the injected water cools the formation, causing thermal contraction. In this study, we numerically investigate thermally induced fractures during water injection using a variational thermo-hydro-mechanical phase-field model. Our simulation results show that cold water injection can nucleate multiple thermal fractures nearly orthogonal to a stimulated fracture, even if the injection pressure is below the fracturing pressure. Further simulation scenarios reveal that thermal fracture propagation is more likely with larger temperature differences, smaller in-situ stress anisotropy, and lower formation permeability. This study highlights the significant impact of thermal effects on fracture initiation and propagation, suggesting the need for careful consideration when regulating or managing fracture initiation during water injection.