In this study, we investigate whether global monopoles cause black holes or wormholes to form. Field equations for static spherically symmetric spacetimes with global monopoles are obtained in f(R) gravity. We found exact solutions for the field equations without using any perturbation or approximation methods. It is shown that the obtained f(R) function is in accordance with the Lambda-cold dark matter (Lambda-CDM) model. Also, it is shown that the static spherically symmetric spacetimes associated with global monopoles form black holes or wormhole structures under some restrictions. Finally, geometrical and physical results of the solutions are discussed.