Movable antennas (MAs) enhance flexibility in beamforming gain and interference suppression by adjusting position within certain areas of the transceivers. In this paper, we propose an MA-assisted integrated sensing and communication framework, wherein MAs are deployed for reconfiguring the channel array responses at both the receiver and transmitter of a base station. Then, we develop an optimization framework aimed at maximizing the sensing signal-to-interference-plus-noise-ratio (SINR) by jointly optimizing the receive beamforming vector, the transmit beamforming matrix, and the positions of MAs while meeting the minimum SINR requirement for each user. To address this nonconvex problem involving complex coupled variables, we devise an alternating optimization-based algorithm that incorporates techniques including the Charnes-Cooper transform, second-order Taylor expansion, and successive convex approximation (SCA). Specifically, the closed form of the received vector and the optimal transmit matrix can be first obtained in each iteration. Subsequently, the solutions for the positions of the transmit and receive MAs are obtained using the SCA method based on the second-order Taylor expansion. The simulation results show that the proposed scheme has significant advantages over the other baseline schemes. In particular, the proposed scheme has the ability to match the performance of the fixed position antenna scheme while utilizing fewer resources.