emailing supervisor to ask about application status I have applied for PhD programme (in Dec 2020) and informed my potential supervisor that I have completed my application. Recently (9 days ago), he asked me if I have received any communication from the institute regarding my application and told me he has not received my application yet. To this, I replied that the Graduate admission office will not answer individual queries regarding application status and I will be receiving emails once the decision has been made (I informed this after making an inquire with the admission office). Also, I had asked him to update me if he has any further updates regarding my application. I have not received any email from the professor after this conversation. The institute has no specific timeline as to when the application decision will be made.  Now, Should I write another email to the professor and ask him if he has received my application? or should I make another inquire in the admission office about my application status by mentioning that the professor had asked me about my application status?
It doesn't seem effective to ask the admissions office again if they have already said that you will hear only when decisions have been made.
But you could give the potential advisor an update that you haven't heard anything yet, asking if he has. But it seems unlikely that one of you would be notified and the other not.
Unless you have some actual reason for needing to know, such as replying about another opportunity, I'd suggest patience. You don't speed anything up by sending pointless emails.