The bandwidth improvement provided by high-bandwidth memory (HBM), and the capability of FPGAs to customize the processing and memory hierarchy, results in a considerable performance increase for memory-intensive work-loads such as graph processing, sorting, machine learning, and database analytics. Modern systems integrating 3D-stacked DRAM memory can be leveraged to realize the Near-Memory Computing (NMC) paradigm by offloading some computations to accelerators placed near the HBM. Although numerous studies have investigated efficient accelerators for FPGA-HBM platforms, researchers have not proposed a systematic way for identifying which application kernels are suitable for execution near the HBM. In this article, we propose compiler support for recognizing offloading candidates without any burden on programmers. Auto-DOK analyzes an application code based on criteria derived from the hardware design goals of FPGA-HBM platforms, and automatically identifies kernels suitable for offloading. We evaluate Auto-DOK on benchmarks ranging from microbenchmarks to real-world kernels. Our results show that Auto-DOK can correctly identify kernels and input sizes suitable for execution near the HBM, and prevents slowdown caused by incorrect offloading decisions for other workloads. Moreover, Auto-DOK operates at compile time with negligible overhead and without the need for expensive profiling.