This paper investigates stability program of Takagi-Sugeno fuzzy systems with interval time-varying delay via a variable delay decomposition approach. By developing a delay decomposition approach, both lower and upper bound information of the delayed plant states can be taken into full consideration; two novel delay-fractional-dependent stability criteria are obtained based on the direct Lyapunov method allied with an appropriate and variable Lyapunov-Krasovskii functional choice and with two different bounding techniques to estimate some integral terms in the time-derivative of the Lyapunov-Krasovskii functional. The first stability criterion is derived by utilizing the suitable and generalized integral inequalities, while the second stability condition is obtained by employing a scalar inequality, without any direct approximation. Particularly, the proposed results differ from previous ones since the positiveness of the Lyapunov-Krasovskii functional is guaranteed by new relaxed conditions. When applying these two stability criteria to check the stability of a T-S fuzzy system, it is shown through some numerical examples that the first stability condition can provide a larger maximum allowable delay bound than the second stability criterion, and both stability criteria yield less conservative than the existing ones.