This paper presents a new extension of the hesitant fuzzy linguistic term set (HFLTS) called intuitionistic fuzzy confidence-based HFLTS that associates an intuitionistic fuzzy value (IFV) with each linguistic term. The resulting term set is termed as intuitionistic fuzzy confidence hesitant fuzzy linguistic term set (IFCHFLTS). The previous studies on the linguistic decision making have emphasized little upon the preference and non-preference for each of the linguistic terms. This information, however, is crucial in multi-criteria decision making under uncertainty. In this regard, we find IFV particularly useful for qualifying each of the linguistic terms with the agent’s degree of preference, non-preference, and hesitation values. Besides, a new aggregation operator named intuitionistic fuzzy confidence linguistic simple weighted geometry (IFCLSWG) is also proposed to fuse decision makers’ linguistic preferences. Further, the criteria weights are estimated using a new method called intuitionistic fuzzy confidence linguistic standard variance. An approach is also suggested for ranking the given alternatives by adapting VIKOR under the proposed IFCHFLTS context. Finally, the practicality and usefulness of the proposal are demonstrated through two real-world problems in green supplier selection for manufacturing industry, and medical diagnosis. The strengths and weaknesses of the proposal are also highlighted by drawing upon a comparison with similar methods.