This paper is concerned with the condition numbers of Gram matrices that arise when computing least square polynomials in polygons of the complex plane. For a stability reason, instead of the power basis \(\{1,\lambda,\lambda^ 2,...,\lambda^ n\}\), polynomials are expressed on the basis of Chebyshev polynomials of the first kind. The author shows that if the polygon is inserted between two ellipses, then the condition number of the \((n+1)\times (n+1)\) Gram matrix is bounded from above by \(4mn(n+1)^ 2(\kappa_ n)^ 2\), where m is the number of edges of the polygon and \(\kappa_ n\geq 1\) is a known ratio such that \(\kappa_ n\) is close to one if the two ellipses are close to each other.