This paper proposes a low-complexity design for the linear weighted MMSE (WMMSE) transmit filters of a coordinated multi-cell system with multiple users per cell. This design is based on a modified WMMSE approach applied to each transmitting base station individually incorporating the signals sent to the cell of interest and the signals leaked to the other cells. This novel design allows for closed-form expressions of the associated Lagrange multipliers and offers a lower computational complexity compared to known methods. In addition, we incorporate this design in an iterative algorithm to find the linear transmit filter maximizing the weighted sum-rate of the multicell system. This algorithm is based on WMMSE where the MSE weights are optimally adjusted so that the WMMSE optimum coincides with the WSR optimum.