A hypersingular integral can be regularized by replacing the whole integrand by a forward difference quotient of 2nd order. If the density function is nearly singular, then Gauss quadrature formulas associated with a suitable modification of the Chebyshev weight function allow to obtain great precision with few nodes. However, in most cases, the own nature of this procedure makes unpredictable the location of quadrature nodes. This paper presents a simple but effective technique whose aim is to mitigate instability when some node lies too close to the pole. Some numerical examples are shown to evaluate the performance of the proposed method.