In the present paper we derive three new and interesting composition formulas for a general class of fractional integral operators involving the product of a general class of polynomials, a general sequence of functions and a multivariable H-function. The operators of our study are quite general in nature and may be considered as extensions of a number of simpler fractional integral operators studied from time to time by several authors. An interesting case of the first composition formula has been given for the sake of illustration. The importance of our study lies in the fact that it unifies and extends a number of corresponding results lying scattered in the literature. In addition, we can also evaluate several double finite integrals with the help of our composition formulas. The results obtained by Buschman [2], Erdélyi [3], Gupta and Soni [7], Goyal and Jain [5], Goyal, Jain and Gaur [6] follow as simple special cases of our composition formulas.