Given a probability measure μ on the unit circle T, consider the reproducing kernel kμ,n(z1,z2) in the space of polynomials of degree at most n-1 with the L2(μ)–inner product. Let u,v∈C. It is known that under mild assumptions on μ near ζ∈T, the ratio kμ,n(ζeu/n,ζev/n)/kμ,n(ζ,ζ) converges to a universal limit S(u, v) as n→∞. We give an estimate for the rate of this convergence for measures μ with finite logarithmic integral. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.