An approach to compute approximate sensitivities in a large eddy simulation (LES) is proposed and assessed.The multi-fidelity sensitivity analysis (MFSA) solves a linearized mean equation, where the mean equation is based on the LES solution. This requires closure modeling which makes the computed sensitivities approximate. The closure modeling is based on inferring the eddy viscosity from the LES data and in predicting the change in turbulence (or the perturbed eddy viscosity) using a simple algebraic model. The method is assessed for the flow over a NACA0012 airfoil at a fixed angle of attack, with the Reynolds number as the varying parameter and the lift, drag, skin friction, and pressure coefficients as the quantities-of-interest. The results show the importance of accurate closure modeling, specifically that treating the eddy viscosity as "frozen" is insufficiently accurate. Also, predictions obtained using the algebraic model for closing the perturbed eddy viscosity are closer to the true sensitivity than results obtained using the fully RANS-based method which is the state-of-the-art and most common method used in industry. The proposed method aims to complement, rather than replace, the current state-of-the-art method in situations in which sensitivities with higher fidelity are required.