The goal of this paper is to develop a high-order numerical method based on the kinetic inviscid flux (KIF) method and flux reconstruction (FR) framework. The KIF aims to find a balance between the excellent merits of the gas-kinetic scheme (GKS) and the lower computational costs. The idea of KIF can be viewed as an inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and Ohwada have made the fundamental contribution. The combination of totally thermalized transport (TTT) scheme and kinetic flux vector splitting (KFVS) method are achieved in KIF. Using a coefficient which is related to time step and averaged collision time, KIF can adjust the weights of TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous splitting, KIF is very suitable and easy to integrate into the existing framework. The well-understood FR framework is used widely for the advantages of robustness, economical costs, and compactness. The combination of KIF and FR is originated by three motivations. The first purpose is to develop a high-order method based on the gas-kinetic theory. The second reason is to keep the advantages of GKS. The last aim is that the designed method should be more efficient. In present work, we use the KIF method to replace the Riemann flux solver applied in the interfaces of elements. The common solution at the interface is computed according to the gas-kinetic theory, which makes the combination of KIF and FR scheme more reasonable and available. The accuracy and performance of present method are validated by several numerical cases. The Taylor-Green vortex problem has been used to verify its potential to simulate turbulent flows.