Solar power innately has issues with weather, grid demand and time of day, which for the case of concentrating solar power (CSP) can be mitigated through use of thermal energy storage. Nuclear reactors, including lead-cooled fast reactors (LFRs), can load follow, but have high fixed and low operating costs which can make this economically unattractive. We investigate potential synergies through coupling CSP and LFR together in a single supercritical CO$_2$ Brayton cycle and/or using the same thermal energy storage. Combining these cycles allows for the LFR to thermally charge the salt storage in the CSP cycle during low demand periods to be dispatched when grid demand increases. The LFR/CSP coupling into one cycle is modeled to find the preferred location of the LFR heat exchanger, CSP heat exchanger, sCO$_2$-to-salt heat exchanger (C2S), turbines, and recuperators within the supercritical CO$_2$ Brayton cycle. Three cycle configurations have been studied: two-cycle configuration, which uses CSP and LFR heat for dedicated turbocompressors, has the highest efficiencies but with less component synergies; a combined cycle with CSP and LFR heat sources in parallel is the simplest with the lowest efficiencies; and a combined cycle with separate high temperature recuperators for both the CSP and LFR is a compromise between efficiency and component synergies. Additionally, four thermal energy storage charging techniques are studied: the turbine positioned before C2S, requiring a high LFR outlet temperature for viability; the turbine after the C2S, reducing turbine inlet temperature and therefore power; the turbine parallel to the C2S producing moderate efficiency; and a dedicated circulator loop. While all configurations have pros and cons, use of a single cycle offers component synergies with limited efficiency penalty. Using a turbine in parallel with the C2S heat exchanger is feasible but results in a low charging efficiency, while a dedicated circulator loop offers flexibility and near perfect heat storage efficiency but increasing cost with additional cycle components.