This paper presents a novel mesh-free approach for solving the Navier–Stokes equations. The method makes use of the meshless method of fundamental solutions (MFS) and the Monte Carlo integration technique for computing the domain integral of the convective terms. No domain or boundary discretization is required. This approach facilitates numerical computation while ensuring accuracy and stability. By imposing a penalty parameter, the Navier–Stokes equations are transformed to resemble the Navier equations of elasticity. Hence, elasticity based fundamental solutions are employed. The proposed formulation is validated through numerical examples, demonstrating its efficacy in capturing steady-state flow phenomena through several examples. This highly parallelized system is then accelerated via GPU computing. Overall, the proposed method provides a promising paradigm for advancing computational fluid mechanics, offering a versatile framework with broad applicability in engineering and scientific domains.