As the digitalization of process industry deepens, process fault detection and diagnosis (FDD) is an essential tool to ensure safe production in chemical industries. However, FDD may have a long detection delay for some chemical faults. Process fault prognosis methods could predict the occurrence of faults in advance, which would give operators more time and reduce the impact of faults. Nevertheless, many fault prognosis methods still suffer from fixed or insufficient prediction time ahead, which greatly confines their usage in critical scenarios. In this paper, we propose a novel Transformer-based multi-variable multi-step (TMM) prediction method for chemical process fault prognosis. Specifically, Transformer models are trained to predict the change of process variables at the next step, and iterative forecasting is used to predict multi-step changes of process variables. Finally, extensive evaluation of applications in a continuous stirred tank heater (CSTH) system and the Tennessee Eastman process (TEP) demonstrates that the proposed TMM prediction method shows high prediction accuracy and early fault prognosis, compared with representative statistical methods and other advanced deep learning methods.