In the nano-scale era, enhancing speed while minimizing power consumption and area is a key objective in integrated circuits. This demand has motivated the development of approximate computing, particularly useful in error-tolerant applications such as multimedia, machine learning, signal processing, and scientific computing. In this research, we present a novel method to create approximate integer multiplier circuits. This work is based on a modification of the well-known Wallace tree multiplier, called the Reduced Complexity Wallace Multiplier (RCWM). Approximation is introduced by replacing conventional Full Adders with approximate ones during the partial product reduction phase. This research investigates the characteristics of 8×8-, 16×16-, and 32×32-bit Approximate Reduced Complexity Wallace Multipliers (ARCWM), evaluating their accuracy, area usage, delay, and power consumption. Given the vast search space created by different combinations and placements of these approximate Adders, a Genetic Algorithm was used to efficiently explore this space and optimize the ARCWMs. The resulting ARCWMs have an area reduction of up to 65% and a power consumption reduction of up to 70%, with no worse delay than the RCWM. Multipliers created with this method can be used in any application that requires parallel multiplication, such as neural accelerators, trading accuracy for area and power reduction. Additionally, an ARCWM can be used alongside a slow shift-and-accumulate multiplier trading off accuracy for faster calculation. This methodology provides valuable guidance for designers in selecting the optimal configuration of approximate Full Adders, tailored to the specific requirements of their applications. Alongside the methodology, we provide all of the tools used to achieve our results as open-source code, including the Register-Transfer Level (RTL) code of the 8×8-, 16×16-, and 32×32-bit Wallace Multipliers.