Sustainable supplier selection has become a strategic activity to enhance the competitiveness of sustainable supply chain management. Research on sustainable supplier selection is considering increasingly more practical factors, such as the uncertainty of decision context and the fuzzy recognition of experts. Evaluation values on different criteria with different characteristic should be represented in their suitable information types to reflect the characteristic accurately and represent experts’ judgments entirely. Moreover, it is difficult, or costly, to build a decision criteria set in which all criteria are independent to each other because of the interaction of technical, economic, environmental and social factors. Therefore, the aim of this paper is to propose a novel decision-making framework for sustainable supplier selection which considers the interaction among criteria with heterogeneous decision information. The proposed framework can not only allow the experts to express their judgments completely, but also improve the efficiency of decision-making. First, a normalized dominance decision matrix based on normalized closeness is built with the heterogeneous decision matrix. Then, a defined discrete Choquet integral multi-criteria distance measure is used to compute the comprehensive associated closeness and rank the alternative sustainable suppliers. This framework provides a new way to handle the interaction among criteria for sustainable supplier selection from the perspective of multi-criteria distance measure, and a novel methodology to solve the problems that the evaluation values cannot be aggregated directly. Finally, an example is given to illustrate the proposed framework for sustainable supplier selection with a comparison analysis.