I applied to several open positions online and so far have received invitations to two different job interviews (company A and B, both ~50 employees). While I'd work for them (currently about to finnish university) and I'm in need of a job, there is another job at company C (> 20k employees) which would be my first choice and out of all three positions I'm arguably matching C's requirements perfectly (as advertised) and my impression is that for me it would be more fun than actual work. So far they only received 3 applications (according to linked in).
I applied 8 days ago to company C but didn't hear anything back except for an automated email response that said they would contact me if they were interested.
I do acknowledge that a week is a short timeframe, but I'm getting in a 2nd round interview with company A and a first round interview with company B this week.
I'd be disappointed to accept an offer of A or B (because I need a job) and hearing back from C a week later. On the other hand, asking only 1 week later might come across as nagging (or even pressuring).
I'd plan to send something like
I'd like to ask about the status of my application because I have received positive feedback from applications elsewhere but your job offering interests me the most and I would really like to grab this opportunity to work in my dream job.