Remaining useful life (RUL) prediction is widely applied in prognostic and health management (PHM) of turbofan engines. Although some of the existing deep learning-based models for RUL prediction of turbofan engines have achieved satisfactory results, there are still some challenges. For example, the spatial features and importance differences hidden in the raw monitoring data are not sufficiently addressed or highlighted. In this paper, a novel multi-head self-Attention fully convolutional network (MSA-FCN) is proposed for predicting the RUL of turbofan engines. MSA-FCN combines a fully convolutional network and multi-head structure, focusing on the degradation correlation among various components of the engine and extracting spatially characteristic degradation representations. Furthermore, by introducing dual multi-head self-attention modules, MSA-FCN can capture the differential contributions of sensor data and extracted degradation representations to RUL prediction, emphasizing key data and representations. The experimental results on the C-MAPSS dataset demonstrate that, under various operating conditions and failure modes, MSA-FCN can effectively predict the RUL of turbofan engines. Compared with 11 mainstream deep neural networks, MSA-FCN achieves competitive advantages in terms of both accuracy and timeliness for RUL prediction, delivering more accurate and reliable forecasts.