Is it acceptable to e-mail a client regarding potential opportunities in their company?

I work as a junior consultant for a medium-sized firm; one of our clients is one of the largest companies in the world, let's call it Company ABC.

One of our contacts in Company ABC, Mr. X, is mid-to-upper management; I have met him and have worked with Mr. X, but always next to a senior consultant/manager from my firm.

During our time together Mr.X has expressed satisfaction with the team and mentioned specifically my understanding of the situation and ability to follow the topic. This happened 3-4 weeks ago, after which he was no longer present and I have now switched projects.

What Mr. X does aligns almost perfectly with my field of studies and my personal and career interests; I am always looking around for potential job opportunities and would like to start looking more seriously in the immediate future.

Is it acceptable to write mr. X an e-mail asking him to consider my profile should he have any openings in his team? What should this e-mail say?

To clarify, I have his e-mail address, but I was always CC'ed to e-mails between him, my manager and my boss. We have never e-mailed each other directly.

EDIT to answer Laconic Droid's comment: it is not a contract issue to go work for a client; in fact, according to some in my field it is supposed to be the "next step" in your career. There is no binding clause in my contract.