Frequency diverse array (FDA) radar exhibits improved beam design flexibility and enhanced spatial-temporal degrees of freedom (DOF) when compared with conventional phased array radar. Due to its unique angle-time/range coupling property, FDA radar has found important applications in target detection. To fully exploit the wide-beam coverage capability of FDA radar, this paper proposes a joint transmit-receive multi-beam design method for FDA radar systems. In the proposed approach, a small number of transmit array elements are utilized to generate wide transmit beams, while a larger number of receive elements are employed to form multiple joint transmit-receive beams with different directions via digital beamforming. This joint design enables simultaneous coverage of multiple directions within the region of interest (ROI), thereby meeting the requirements of wide-area search tasks in radar applications. The high degrees of freedom offered by the receive array not only enhance the flexibility of joint beamforming but also contribute to the reduction of the mainlobe width of the joint transmit-receive beam. In addition, a Taylor window-based non-uniform weighting scheme is introduced at the receiver to suppress beam sidelobes. The simulation outcomes substantiate the efficacy of the proposed approach.