It's certainly possible to create AI systems that ask questions. Various forms of expert systems and diagnostic support system applications already do that. As to the question of whether or not they are curious, that's one I'll leave to the philosophers. But it is absolutely possible to create an AI that attempts to reason out a solution to a problem, find that it is unable to generate an acceptable answer, and then prompts the user for more information.
One context where this can be done is Abductive Inference systems for medical decision making. I'd refer you to Abductive Inference Models for Diagnostic Problem Solving by Reggia and Peng, or Computer Assisted Medical Decision Making 1 by Reggia and Tuhrim for more on that specific point.