The Asymmetric Distance-Constrained Vehicle Routing Problem (ADVRP) is an NP-hard problems. In ADVRP problem, each customer is visited once by one vehicle; every tour starts and ends at a depot; and the travelled distance by each vehicle is required to be less than or equal to the given maximum value. The problem is a natural extension of Vehicle Routing Problem case. In our work, we propose a hybrid metaheuristic algorithm combining the Randomized Variable Neighborhood Search (RVNS) and the Tabu Search (TS) to solve the problem. The combination of multiple neighborhoods and tabu mechanism is used for their capacity to escape local optima while exploring the solution space. Furthermore, the intensification and diversification phases are also included to deliver optimized and diversified solutions for the search. Extensive numerical experiments on benchmark instances show that our algorithm can be comparable with the state-of-the-art previous algorithms in terms of solution quality and computation time. In many cases our proposed method is able to improve the best-known solution available from the literature.