The realizations of inequalities involving fractional integral and differential operators are considered to be important due to their wide range of applications among researchers. In this work, we introduce new fractional integral inequalities of Minkowski’s type using the Riemann–Liouville fractional integral operator. We replace the constants appearing in Minkowski’s inequality with two positive functions. Furthermore, we establish new fractional inequalities related to reverse Minkowski-type inequalities via the Riemann–Liouville fractional integral. Using this fractional integral operator, some special cases of reverse Minkowski-type inequalities are also discussed.