In spoken languages, utterances are often shaped to be incomplete or vague for efficiency. This can lead to varying interpretations of the same input, based on different assumptions about the context. To ensure reliable user-model interactions in such scenarios, it is crucial for models to adeptly handle the inherent ambiguity in user queries. However, conversational agents built upon even the most recent large language models (LLMs) face challenges in processing ambiguous inputs, primarily due to the following two hurdles: (1) LLMs are not directly trained to handle inputs that are too ambiguous to be properly managed; (2) the degree of ambiguity in an input can vary according to the intrinsic knowledge of the LLMs, which is difficult to investigate. To address these issues, this paper proposes a method to align LLMs to explicitly handle ambiguous inputs. Specifically, we introduce a proxy task that guides LLMs to utilize their intrinsic knowledge to self-disambiguate a given input. We quantify the information gain from the disambiguation procedure as a measure of the extent to which the models perceive their inputs as ambiguous. This measure serves as a cue for selecting samples deemed ambiguous from the models' perspectives, which are then utilized for alignment. Experimental results from several question-answering datasets demonstrate that the LLMs fine-tuned with our approach are capable of handling ambiguous inputs while still performing competitively on clear questions within the task.