I'm applying to graduate school and there is a professor at my home university I am interested in working with. As a bit of background, I have been doing undergraduate research with this professor for about a year now and he seems to like me (though maybe he's nice to everyone. I can't be certain). My research under him has been going well as hes told me I've been doing good work. In fact, he uses exclamation marks sometimes, which I find to be odd. Fast forwarding a bit, I've recently wrapped up the project. Given my academic history with him, if he was interested in working with me, would he ask me to be his student (sorry if this comes off as having an inflated ego, I don't have one.) All in all, I'm not sure how to proceed in asking him.
Just ask. There is nothing to lose. Many of us would be hesitant to ask a student, realizing that there are advantages to moving on and coming in contact with different faculty and their ideas. He might even suggest that you move on, actually, and don't take that as a negative.
If that happens, ask for his advice on your next steps. And try to keep in touch with him over the next years. An email every year might be welcome if you are progressing nicely.
These things are best done in person if that is an option.
But think about whether this prof (the easy option) or moving on is the best option for your future.
The formalities of admission depend from geography to geography, and institution to institution.
However, it is most emphatically up to you to take the initiative to indicate interest and spur exploration of whether this would make sense. I would have thought it entitled, arrogant, and forward as a faculty member to indicate to a student, even a close collaborator, unsolicited, that I would be happy to be their advisor!
The most I would have felt comfortable is suggesting to a student that I'd be delighted to discuss their future plans with them if they'd like. And I'd probably assume the best I could do for them is help them navigate their choices of graduate school and introduce them to my collegues elsewhere.
Good luck. And as others have said, be open to the idea that the best for you might be to retain this collaborative relationship informally and continue your studies elsewhere.