Traditionally, the High-Level Synthesis (HLS) for Field Programmable Gate Array (FPGA) devices is a methodology that transforms a behavioral description, as the timing-independent specification, to an abstraction level that is synthesizable, like the Register Transfer Level. This process can be performed under a framework that is known as Design Space Exploration (DSE), which helps to determine the best design by addressing scheduling, allocation, and binding problems, all three of which are NP-hard problems. In this manner, and due to the increased complexity of modern digital circuit designs and concerns regarding the capacity of the FPGAs, designers are proposing novel HLS techniques capable of performing automatic optimization. HLS has several conflicting metrics or objective functions, such as delay, area, power, wire length, digital noise, reliability, and security. For this reason, it is suitable to apply Multiobjective Optimization Algorithms (MOAs), which can handle the different trade-offs among the objective functions. During the last two decades, several MOAs have been applied to solve this problem. This paper introduces a comprehensive analysis of different MOAs that are suitable to perform HLS for FPGA devices. We highlight significant aspects of MOAs, namely, optimization methods, intermediate structures where the optimizations are performed, HLS techniques that are addressed, and benchmarks and performance assessments employed for experimentation. In addition, we show the analysis of how multiple objectives are optimized currently in the algorithms and which are the objective functions that are optimized. Finally, we provide insights and suggestions to contribute to the solution of major research challenges in this area.