*direct* interactions mediated through flows that are generated through the actuation of the filaments themselves, and *indirect* interactions mediated through the motion of the cell body (i.e., flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a minimal singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction in the entire geometric parameter range; (ii) that for both thrust- and rotation-induced flows indirect advection dominates for long filaments and at wide separation, i.e., primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other’s wake, which we characterize mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analyzing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is nonmonotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.