© 2019 American Physical Society.Out-of-time-order correlators (OTOCs) play an increasingly important role in different fields of physics and in particular they provide a way of quantifying information scrambling in quantum many-body systems. We verify that an OTOC can be used to probe an excited-state quantum phase transition (ESQPT) in a quantum many-body system. We examine the dynamical properties of an OTOC in the Lipkin-Meshkov-Glick model, which undergoes an ESQPT, using the exact diagonalization method. We show that the long-time evolution of the proposed OTOC is remarkably different in the different phases of the ESQPT. In consequence, we put the long-time averaged value of the OTOC forward as a possible ESQPT order parameter. Our results highlight the connections between OTOCs and ESQPTs, opening the possibility of using OTOCs for accessing experimentally ESQPTs in quantum many-body systems.