•We propose a linguistic GDM approach based on linguistic distributions and HFLTSs.•Our model aims at maximizing the support degree of the group opinion.•The accuracy of the group opinion in our model is guaranteed.•A mixed 0–1 linear programming approach is presented to solve our model.•The use of our model in MAGDM is demonstrated. The hesitant fuzzy linguistic term set (HFLTS) and the linguistic distribution are becoming popular tools to model linguistic expressions with multiple linguistic terms in decision problems. Compared with HFLTSs, linguistic distributions provide more probabilistic preference information over linguistic terms, and are useful to express decision makers' preferences accurately. However, in a group decision context a linguistic distribution based group opinion will bring great difficulty for the group to take an accurate action. Meanwhile, the linguistic group opinion should obtain enough support from decision makers in the group. To tackle these issues, based on the use of linguistic distributions and HFLTSs we propose a new linguistic group decision model called the maximum support degree model (MSDM), aiming at maximizing the support degree of the group opinion as well as guarantying the accuracy of the group opinion. A mixed 0–1 linear programming approach is presented to solve the MSDM, and a feedback adjustment is employed to improve the support degree of the group opinion. Finally, the use of the MSDM in multiple attribute group decision making is demonstrated.