Well, you'll find out just how eager they are to sell.
If your Realtor said it was a good asking price, did he run the comps on the property? Did he show them to you? You really ought to see the data, too.
But assuming that $193,500 is a good price already, if they're "very eager" to sell they may well take your lower offer. Maybe they would have gone even lower. Very eager? You only offered 5% under list. Now, if it's the deal of the century, and they know it, then maybe they won't be too flexible. The real question is: What's the property worth?
So, see what happens. Their response will reveal just how eager they truly are.