The Cryogenic Underground Observatory for Rare Events (CUORE) is designed to search for neutrinoless double beta decay of 130Te with an array of 988 TeO₂ bolometers operating at temperatures around 10 mK. The experiment is currently being commissioned in Hall A of Laboratori Nazionali del Gran Sasso, Italy. The goal of CUORE is to reach a 90% C.L. exclusion sensitivity on the ¹³⁰Te decay half-life of 9 × 10²⁵ years after 5 years of data taking. The main issue to be addressed to accomplish this aim is the rate of background events in the region of interest, which must not be higher than 10⁻² counts/keV/kg/y. We developed a detailed Monte Carlo simulation, based on results from a campaign of material screening, radioassays, and bolometric measurements, to evaluate the expected background. This was used over the years to guide the construction strategies of the experiment and we use it here to project a background model for CUORE. In this paper we report the results of our study and our expectations for the background rate in the energy region where the peak signature of neutrinoless double beta decay of ¹³⁰Te is expected.