Negatively charged nanoparticles (NPs) in 1:1, 1:2, and 1:3 electrolyte solutions are studied in a primitive ion model using molecular dynamics (MD) simulations and classical density functional theory (DFT). We determine the conditions for attractive interactions between the like-charged NPs. Ion density profiles and NP-NP interaction free energies are compared between the two methods and are found to be in qualitative agreement. The NP interaction free energy is purely repulsive for monovalent counterions, but can be attractive for divalent and trivalent counterions. Using DFT, the NP interaction free energy for different NP diameters and charges is calculated. The depth and location of the minimum in the interaction depend strongly on the NPs' charge. For certain parameters, the depth of the attractive well can reach 8-10 kBT, indicating that kinetic arrest and aggregation of the NPs due to electrostatic interactions is possible. Rich behavior arises from the geometric constraints of counterion packing at the NP surface. Layering of counterions around the NPs is observed and, as secondary counterion layers form the minimum of the NP-NP interaction free energy shifts to larger separation, and the depth of the free energy minimum varies dramatically. We find that attractive interactions occur with and without NP overcharging.