It's not allowed under the current system. See Bounty-like facility for rewarding excellent questions and any of its duplicates for the rationale.
The problem with offering bounties on questions is it doesn't encourage any behavior from other users. When users see a bounty available on a question, it encourages them to provide a detailed answer in an effort to be rewarded the bounty.
You could set a bounty on behalf of the user to help him/her get a good answer on the question. Another (related) option would be to judiciously share a link to the question elsewhere to draw attention to it.
You could technically award a bounty on one of the user's answers, but consider whether this is really in the spirit of the system to do so.