The dynamics of globular clusters has been studied in great detail in the context of general relativity as well as with modifications of gravity that strongly depart from the standard paradigm such as Modified Newtonian Dynamics. However, at present there are no studies that aim to test the impact that less extreme modifications of gravity (e.g. models constructed as alternatives to dark energy) have on the behaviour of globular clusters. This Letter presents fits to the velocity dispersion profile of the cluster NGC 2419 under the symmetron-modified gravity model. The data show an increase in the velocity dispersion towards the centre of the cluster which could be difficult to explain within general relativity. By finding the best-fitting solution associated with the symmetron model, we show that this tension does not exist in modified gravity. However, the best-fitting parameters give a model that is inconsistent with the dynamics of the Solar system. Exploration of different screening mechanisms should give us the chance to understand if it is possible to maintain the appealing properties of the symmetron model when it comes to globular clusters and at the same time recover the Solar system dynamics properly.