We give a local search based algorithm for k-median (k-means) clustering from the perspective of individual fairness. More precisely, for a point x in a point set P of size n, let r(x) be the minimum radius such that the ball of radius r(x) centered at x has at least n/k points from P. Intuitively, if a set of k random points are chosen from P as centers, every point x∈ P expects to have a center within radius r(x). An individually fair clustering provides such a guarantee for every point x∈ P. This notion of fairness was introduced in [Jung et al., 2019] where they showed how to get an approximately feasible k-clustering with respect to this fairness condition. In this work, we show how to get an approximately optimal such fair k-clustering. The k-median (k-means) cost of our solution is within a constant factor of the cost of an optimal fair k-clustering, and our solution approximately satisfies the fairness condition (also within a constant factor).