The Birkhoff conjecture says the boundary of a strictly convex integrable billiard table is necessarily an ellipse. Here we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. One of the crucial ideas in the proof consists in analyzing expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter near the origin, deriving and studying a higher order condition. This is a joint work with Vadim Kaloshin and Alfonso Sorrentino.