The dynamical evolution of self-interacting scalars is of paramount importance in cosmological settings and can teach us about the content of Einstein's equations. In flat space, nonlinear scalar field theories can give rise to localized, nonsingular, time-dependent, long-lived solutions called "oscillons." Here, we discuss the effects of gravity on the properties and formation of these structures, described by a scalar field with a double well potential. We show that oscillons continue to exist even when gravity is turned on, and we conjecture that there exists a sequence of critical solutions with an infinite lifetime. Our results suggest that a new type of critical behavior appears in this theory, characterized by modulations of the lifetime of the oscillon around the scaling law and the modulations of the amplitude of the critical solutions.