In this article, we focus on the stability of asynchronous sampled-data interconnected systems and the evaluation of how asynchronous sampling and time delays degrade the $\mathcal {L}_{2}$ gains from exogenous inputs to controlled outputs. To judge the stability and evaluate the performance of systems, an integral approach is proposed and it combines a revised form of input-to-state stability Lyapunov conditions with an integral description of bounded measurement errors. Practical procedures for the application of the approach are further presented. In the case with shared sampling periods, a time-translation approach is introduced to investigate the relationship between asynchronous sampling and time delays.