In this paper, we propose a new transmission scheme, named as Integer Forcing-and-Forward (IFF), for communications among multi-pair multiple-antenna users in which each pair exchanges their messages with the help of a single multi antennas relay in the multiple-access and broadcast phases. The proposed scheme utilizes Integer Forcing Linear Receiver (IFLR) at relay, which uses equations, i.e., linear integer-combinations of messages, to harness the intra-pair interference. Accordingly, we propose the design of mean squared error (MSE) based transceiver, including precoder and projection matrices for the relay and users, assuming that the perfect channel state information (CSI) is available. In this regards, in the multiple-access phase, we introduce two new MSE criteria for the related precoding and filter designs, i.e., the sum of the equations MSE (Sum-Equation MSE) and the maximum of the equations MSE (Max-Equation MSE), to exploit the equations in the relay. In addition, the convergence of the proposed criteria is proven as well. Moreover, in the broadcast phase, we use the two traditional MSE criteria, i.e. the sum of the users' mean squred errors (Sum MSE) and the maximum of the users' mean squared errors (Max MSE), to design the related precoding and filters for recovering relay's equations by the users. Then, we consider a more practical scenario with imperfect CSI. For this case, IFLR receiver is modified, and another transceiver design is proposed, which take into account the effect of channels estimation error. We evaluate the performance of our proposed strategy and compare the results with the conventional amplify-and-forward (AF) and denoise-and-forward (DF) strategies for the same scenario. The results indicate the substantial superiority of the proposed strategy in terms of the outage probability and the sum rate.