The ability of reasoning beyond data fitting is substantial to deep learning systems in order to make a leap forward towards artificial general intelligence. A lot of efforts have been made to model neural-based reasoning as an iterative decision-making process based on recurrent networks and reinforcement learning. Instead, inspired by the consciousness prior proposed by Yoshua Bengio, we explore reasoning with the notion of attentive awareness from a cognitive perspective, and formulate it in the form of attentive message passing on graphs, called neural consciousness flow (NeuCFlow). Aiming to bridge the gap between deep learning systems and reasoning, we propose an attentive computation framework with a three-layer architecture, which consists of an unconsciousness flow layer, a consciousness flow layer, and an attention flow layer. We implement the NeuCFlow model with graph neural networks (GNNs) and conditional transition matrices. Our attentive computation greatly reduces the complexity of vanilla GNN-based methods, capable of running on large-scale graphs. We validate our model for knowledge graph reasoning by solving a series of knowledge base completion (KBC) tasks. The experimental results show NeuCFlow significantly outperforms previous state-of-the-art KBC methods, including the embedding-based and the path-based. The reproducible code can be found by the link below.