Field-programmable gate arrays (FPGAs) offer the inherent ability to reconfigure at runtime, making them ideal for applications such as data centers, cloud computing, and edge computing. This reconfiguration, often achieved through remote access, enables efficient resource utilization but also introduces critical security vulnerabilities. An adversary could exploit this access to insert a dormant hardware trojan (HT) into the configuration bitstream, bypassing conventional security and verification measures. To address this security threat, we propose a supervised learning approach using deep recurrent neural networks (RNNs) for HT detection within FPGA configuration bitstreams. We explore two RNN architectures: basic RNN and long short-term memory (LSTM) networks. Our proposed method analyzes bitstream patterns, to identify anomalies indicative of malicious modifications. We evaluated the effectiveness on ISCAS 85 benchmark circuits of varying sizes and topologies, implemented on a Xilinx Artix-7 FPGA. The experimental results revealed that the basic RNN model showed lower accuracy in identifying HT-compromised bitstreams for most circuits. In contrast, the LSTM model achieved a significantly higher average accuracy of 93.5%. These results demonstrate that the LSTM model is more successful for HT detection in FPGA bitstreams. This research paves the way for using RNN architectures for HT detection in FPGAs, eliminating the need for time-consuming and resource-intensive reverse engineering or performance-degrading bitstream conversions.