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This manuscript proposes a multi-scale link prediction approach based on Graph Neural Networks (GNNs). The proposed method - Multi-Scale Line-Graph Link Prediction (MLGLP) - learns the graph structure and extracts effective representative features of graph edges to address challenges such as information loss and handle multi-scale information. This approach utilizes embedding vectors generated by GNNs from enclosing subgraphs. While expanding GNN layers can capture more intricate relations, it often leads to overs-smoothing. To mitigate this issue, we propose constructing coarse-grained graphs at three distinct scales to uncover complex relations. To apply multi-scale subgraphs in GNNs without using pooling layers that lead to information loss, we convert each subgraph into a line-graph and reformulate the task as a node classification problem. The hierarchical structure facilitates exploration across various levels of abstraction, fostering deeper comprehension of the relationships and dependencies inherent within the graph. The proposed method is applied on link prediction problem, which can be modelled as a graph classification problem. We perform extensive experiments on several well-known benchmarks and compare the results with state-of-the-art link prediction methods. The experimental results demonstrate the superiority of our proposed model in terms of average precision and area under the curve.