The paper is concerned with the stability of linear systems with interval time-varying delay. Through constructing a new augmented Lyapunov-Krasovskii functional (LKF) which contains some quadruple-integral terms and estimating the time derivative of the LKF less conservatively, new stability criteria are derived without introducing any free matrices. Moreover, by proving the positive definiteness of the LKF with some integral inequalities, the constraints on some functional parameters are relaxed and the conservatism of the obtained results are further reduced. Numerical examples are also given to demonstrate the effectiveness and reduced conservatism of the obtained results.