At present, soft real-time data center applications are in a booming development and impose stringent delay requirements on internal data transfers. In this context, many recently proposed data center transport protocols share a common goal of minimizing Flow Completion Time (FCT), and the Shortest Remaining Processing Time (SRPT) scheduling algorithm has attracted widespread attentions for its superior performance in average FCT. However, SRPT suffers from the instability problem, incurring more and more flows left uncompleted even if the traffic load is within the fabric capacity, which implies unnecessary bandwidth waste. To solve the problem, this paper proposes a backlog-aware flow scheduling algorithm (BASRPT) for both giant switch and general topologies. Because of taking into account queue backlogs other than flow sizes at scheduling, we prove that BASRPT is stable and still maintains good FCT performance. To overcome the huge computation overhead and enable distributed implementation, a fast and practical approximation algorithm called fast BASRPT is also developed. Extensive flow-level simulations show that fast BASRPT indeed stabilizes the queue length and obtains a higher throughput while being able to push the FCT arbitrarily close to the optimal value in the condition of feasible traffic loads.