When we did it, it was at the very beginning. After the showing, we called back to say that yes, we were very interested, but we were hoping to keep the rent at $xxxx, which seemed in line with the current market. Would they be open to that? (It was shown by an agent, not the owner.) A few hours later he called back and said that amount would be fine, and we then submitted our app. (It ended up falling through because the house had been listed for both rent and sale, and they got a purchase offer before we signed the lease, but that seemed entirely unrelated.)
After that fell through, we looked at our current place. Price on this one seemed fair-ish and we were getting desperate, but when we called back (talking to the owner) to say that we were definitely interested, we asked about a 3 year lease (instead of 1). I was anticipating him saying that he'd do it was an increase of $50 or $100 per year, which is what we'd heard from other places, but he did it with no increase at all. (crazy, IMO, in this market). But again, that was done before we submitted the application.
So I guess I'd call back and say that we are extremely interested but that based on comps, $2700 seems more in line with the market and would be the willing to sign a lease at that amount? They either say yes, or no. If they say no, you likely haven't lost the place. You can always go up to asking at that point. It's unlikely they would hold the negotiating against you so much that they would reject an otherwise attractive offer (assuming everything else about your offer is strong). I wouldn't mention that you just want to get into the neighborhood, because that sounds like, "I will move when my lease is up, so this is a one year thing," which would make me less than enthused as a landlord.