This paper discusses the regularity of stochastic convolution of a class of linear stochastic functional differential equations with distributed delay driven by fractional Brownian motion. In this paper, the regularity of stochastic convolution is obtained by using the estimation of fundamental solutions and the properties of fractional Brownian motion. In particular, when the coefficient operator of the equation conforms to some specific conditions, this paper gives a condition satisfying the regularity of stochastic convolution.