Contacting professors for PhD supervision past the formal application deadline of the university in USA
In the US this would certainly be unusual and at many universities, impossible. Admission to doctoral programs is done by a committee and there are cost implications for admitting a student. Space is needed, if nothing else. An individual professor, most places, has little control over that. 

You can ask, of course, and there is no issue with exploring options with the professor. 

But in the US there is more than research involved in a doctorate. There are, at least, qualifying exams. And most students require advanced coursework in order to pass the exams. This normally occurs before the serious dissertation research begins, or at the earliest stages of exploration. Many students don't choose an advisor until they pass comps, since failure means you will be asked to leave. 

But, another option, provided that you are good enough is to state your case and show that you are a worthy "independent researcher" who wants to work with the professor now on some ideas and that you will be applying later to the degree program. Some professors might find that interesting, others not. It might depend on where they are in their own career and how generally busy they are with their normal duties. 

And, I'm going to guess that you aren't alone in this wish and that your query might just get lost in the mass of others. Think of a "renowned professor" as a kind of magnet for such interest.