I have spoken with a few people who are in their OPT, and who have told me that they are uncomfortable with bringing up, with their boss, the topic of getting H-1B status. If you feel this way, here are some thoughts and suggestions. Ihave never had to ask for H-1B status, but believe, based on my experience in consulting about and teaching classes on communication, that my suggestions will be helpful.
First, remember Sun Tzu, and his advice, from "The Art of War," about knowing the enemy and knowing yourself. In other words, look at the situation from your boss's perspective. In particular, ask yourself how your boss and your company would benefit from petitioning for H-1B status for you. Even though you yourself would benefit, that idea should NOT be the basis for talking to your boss. In other words, avoid talking about why you yourself would benefit from H-1B. Simply put, your boss probably doesn't care.
Second, try to get away from thinking that H-1B is a matter of "you versus your boss." Instead, think of the issue as being one where you and your boss have a common objective, i.e. the need to solve an immigration problem--namely, that unless something happens, you will need to stop working at a certain point in the future, namely, when your OPT ends.
So if we combine these two ideas, we can bring up H-1B in the context of alerting our boss to a potential problem. The boss needs to realize that we might not be there in the near future. On the other hand, if your continued presence is something the boss wants, then you can discuss how to resolve the issue--namely, via H-1B status. By getting this status, the boss avoids disruption in work, continues to have someone to work with clients or customers and gets to keep a valuable employee. You also look like a genius to your boss, in that you have identified a problem, but also have a way to solve it, thereby making your boss's life easier.
This previous paragraph illustrates my third point: do not think, by discussing H-1B, that you are asking for charity or a handout from your boss. Think of yourself as someone who is bringing value to your company and to your boss. Therefore, the decision about H-1B, from your boss's view, should simply make good business sense, because the cost of doing it is far outweighed by the value you bring.
If your boss is American, you can try the "bad news / good news" approach, which is a common form of American humor, and one therefore that your boss will recognize. Your conversation might go like this: "Boss, I have bad news and I have good news. The bad news is that unless we take action, I will need to leave this job on or before [date], due to immigration requirements. The good news is that I know the action we need to take to resolve the problem."
I hope these suggestions are helpful to you.
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