Abstract: We derive geometrical bounds on the irreversibility for both classical and open quantum Markovian systems that satisfy the detailed balance conditions. Using the information geometry, we prove that the irreversible entropy production is bounded from below by a modified Wasserstein distance between the initial and final states, thus generalizing the Clausius inequality. The modified Wasserstein metric can be regarded as a discrete-state generalization of the Wasserstein metric, which plays an important role in the optimal transport theory. Notably, the derived bounds can be interpreted as classical and quantum speed limits, implying that the associated entropy production constrains the minimum time required to transform a system state. We illustrate the results on several systems and demonstrate that a tighter bound on the efficiency of quantum heat engines can be obtained.