No need to ask until you know you have other offer waiting
You work there already, so they have a strong reason to inform you and your boss about their decision. If they didn't, then either there is no decision or they don't want you to know it, for example to prevent loss of enthusiasm if they are planning to let you go. In both situations asking makes no sense, there is nothing to tell you or they don't want to do it. And you are not losing anything by not asking at this point.
Asking your boss / manager during informal conversation shouldn't usually be a problem, but if you have a reason to believe he might see it as "disrespectful and arrogant", then it might be safer not to do it. If these are your concerns, and your boss never gave you any reason for them, then don't worry about it too much.
Consider approaching him in an informal setting (like cafeteria) and asking something like this:
Do you know how long it will take HR to give me some feedback?
As you are aware, my contract ends in x time, and I had an interview in regards to this on x day, since I would very much like to continue working here. When can I expect an answer on the issue so I can plan ahead for next steps? wording by Leon
That's it. Nothing rude, disrespectful or arrogant. Just signaling you are still waiting.
If you have better job offer
Just take it, and inform your current company that you are no longer interested, thank you, it was great to work with you, good bye.
If you have an acceptable job offer but would rather stay at your current company
This is the only time asking is something you have to do, even if your manager might be a bit over-sensitive about it or tries avoid this conversation. Inform your boss that you have another job offer, that you would prefer to stay with your current company but first things first, you need employment and this is the offer, you have X days to sign it so your current company needs to give you an offer in X-2 days (or earlier) for you to be able to consider it. Ask your manager what he thinks is the best course of action in this situation. Be calm, professional, don't accuse your company of being slow or anything. Just state the facts as you know them.
Boss, I'm sorry but I need to know what the decision about my employment will be. I have other offers, I need to sign one of them before [insert date here] if I want it. I would prefer to stay here, but I can't afford the risk of not being employed neither here nor there, I hope you understand. Should I talk to HR? Or accept one of the other offers? Because if [current company] will not give me an offer before [insert a bit earlier date here] I won't have a choice, I will have to work for someone else. What do you think?