The integrated sensing and communication (ISAC) technology has been extensively researched to enhance communication rates and radar sensing capabilities. Additionally, a new technology known as fluid antenna system (FAS) has recently been proposed to obtain higher communication rates for future wireless networks by dynamically altering the antenna position to obtain a more favorable channel condition. The application of the FAS technology in ISAC scenarios holds significant research potential. In this paper, we investigate a FAS-assisted multiple-input multiple-output (MIMO) ISAC system for maximizing the radar sensing signal-clutter-noise ratio (SCNR) under communication signal-to-interference-plus-noise ratio (SINR) and antenna position constraints. We devise an iterative algorithm that tackles the optimization problem by maximizing a lower bound of SCNR with respect to the transmit precoding matrix and the antenna position. By addressing the non-convexity of the problem through this iterative approach, our method significantly improves the SCNR. Our simulation results demonstrate that the proposed scheme achieves a higher SCNR compared to the baselines.