This study examines the transition to turbulence downstream of fluttering and non-fluttering bioprosthetic aortic valves using global linear stability theory. During systole, increasing inflow velocities result in temporally evolving flow profiles downstream of the valve which are highly influenced by the leaflet kinematics. These profiles are time averaged at the sinotubular junction over successive windows and used as boundary conditions to obtain base flows for stability analysis. Three-dimensional global modes are computed for one design of each valve type across multiple time windows, revealing several unstable modes whose frequencies and growth rates increase over time. Notably, the non-fluttering valve exhibits higher growth rates than the fluttering valve. The resulting eigenspectra show that, for each case, the most unstable eigenvalues align along two distinct parabolic branches in the complex plane. For each valve case, the modes within each branch are found to have similar group velocities, suggesting that the unstable modes along a branch constitute a coherent structure. Motivated by this, a transient growth analysis is conducted to identify the optimal initial perturbations that maximise energy gain for a given time horizon. When superimposed onto the base flow, these perturbations generate vortical structures that closely resemble those observed in fully coupled nonlinear fluid–structure interaction simulations for a similar time scale as the one used to obtain the optimal perturbations. These results suggest that the optimal perturbations may initiate the shear-layer instabilities responsible for transition to turbulence, providing valuable insight into the underlying mechanisms in the flow fields downstream of bioprosthetic valve designs.