In this article, a class of decentralized approximate Newton (DEAN) methods for addressing convex optimization on a networked system is developed, where nodes in the networked system seek a consensus that minimizes the sum of their individual objective functions through local interactions only. The proposed DEAN algorithms allow each node to repeatedly perform a local approximate Newton update, which leverages tracking the global Newton direction and dissipating the discrepancies among the nodes. Under less restrictive problem assumptions in comparison with most existing second-order methods, the DEAN algorithms enable the nodes to reach a consensus that can be arbitrarily close to the optimum. Moreover, for a particular DEAN algorithm, the nodes linearly converge to a common suboptimal solution with an explicit error bound. Finally, simulations demonstrate the competitive performance of DEAN in convergence speed, accuracy, and efficiency.