To describe the linguistic neutrosophic information with uncertain variables in the complex decision-making problems, this paper originally defines the concept of a linguistic neutrosophic uncertain number (LNUN). The LNUN consists of three uncertain linguistic variables given by three neutrosophic linguistic numbers to describe the truth, falsity, and indeterminacy linguistic information independently. Then, both the basic operational laws of LNUNs and the score and accuracy functions of LNUNs are presented for the ranking of LNUNs. After that, two operators including a LNUN weighted arithmetic averaging (LNUNWAA) operator and a LNUN weighted geometric averaging (LNUNWGA) operator are developed to aggregate LNUN information. Based on the aggregation operators, a novel multiple attribute group decision-making (MAGDM) method is established under a LNUN environment. Finally, an example of investment decision is illustrated to demonstrate the application and the effectiveness of the developed method.