The Secure Hash Algorithm-256 (SHA-256) is a cryptographic function used in a wide variety of applications ranging from Internet of Things micro-devices to highperformance systems. This paper studies a set of implementations of the SHA-256 on a field-programmable gate array (FPGA) elaborated using the Open Computing Language (OpenCL). These implementations apply several optimization techniques to improve their respective throughputs. Reported results show that a combination of OpenCL optimization techniques allows obtaining an implementation offering a 90x speed-up when compared to an unoptimized OpenCL implementation. Moreover, the best reported optimized implementation achieves a throughput of 3973 Mbps, which is 4.3 times higher than the best previously published HLS-based SHA-256 implementation and even higher than the previously published implementations using a hardware description language. To our knowledge, this work is the first that proposes an OpenCL-based FPGA implementation of SHA-256 and its OpenCL-based optimization methodology.