For a reactor to establish a sustainable breed-and-burn (B&B) mode of operation, its fuel has to reach a minimum level of average burnup. The value of the minimum required average discharge burnup strongly depends on the core design details. Using the extended neutron balance method, it is possible to quantify the impact of major core design choices on the minimum required burnup in a B&B core. Relevant design variables include the fuel chemical form, nonactinide mass fraction of metallic fuel, feed-fuel fissile fraction, fuel rod pitch-to-diameter ratio (P/D), average neutron flux level, and fraction of neutron loss. Metallic fuels have been found to be the only viable fuel options for a realistic near-term B&B reactor. For the core designs we have studied, it was not possible to sustain B&B operation using oxide fuel that is not enriched, while nitride and carbide fuels may only work in highly ideal low-leakage systems at very high levels of discharge burnup and, hence, neutron irradiation damage. The minimum required burnup increases strongly with the total fraction of neutrons that is lost to leakage and reactivity control. The flux level has no effect on the neutron balance within the applicable range, and the average discharge burnup is also relatively insensitive to the fraction of fissile material in the feed fuel in the range from depleted uranium (0.2% 235U) to natural uranium (0.71% 235U). The minimum required burnup is most sensitive, in order of importance, to the fractional loss of neutrons, the Zr content in metallic fuel, and the fuel rod P/D. Changing the weight fraction of zirconium in metallic fuel by 1% (for example, from 10% to 9%) gives the same change in required discharge burnup as adjusting the P/D by 0.02 (for example, from 1.10 to 1.12).