"Pushing" a search committee for an answer? I had a job interview (Skype) for a post-doc at university A last Thursday, which I think went well. They say they would take 1 week to 10 days to get back to me.

I had applied to other positions before, and got a much less attractive offer from  university B. I just received an email from university B saying that they'd really need to close the hiring process by the end of this week and would expect an answer from me in the next couple of days.

Would it be too aggressive to send an e-mail to University A to inquire about my chances when it's only been a few days after the interview? On the other hand, it is pretty late in the job hunting season so I feel it would be understandable?
No, it’s not too aggressive. A very common thing to do in your situation is to inform university A of your deadline with university B (and to mention to them that you prefer university A).