Discontinuities are natural structures that exist in rocks and can affect the stability of rock structures. In this article, the influence of notch presence on the failure evolution around a hole in compressed rock specimens is investigated numerically. Firstly, the uniaxial compressive test on a rock specimen with a circular hole is modeled and the failure evolution in the specimen is simulated. In a separate model, notches are created at the surface of the hole. Results show that when the notches are created in the model, failure zone around the hole is transferred to a distance away from the surface of the hole. In addition, a parametric study is carried out to investigate the influence of the notch length and the confining pressure on the fracturing behavior of the specimen. Numerical results presented in this article indicate that the presence of notches at the surface of the hole and their dimensions can affect the fracturing mechanism of the specimen. In some cases, the failure at the boundary of the hole is prevented when the notches of certain dimensions are added to the hole. The insights gained from this numerical study may be helpful to control the failure around underground excavations.