Why does your agent feel as if the property is worth closer to $260? Does what he says make sense?
First of all, be very careful about comps. You should see for yourself what's out there. Obviously you can't see what sold, although you can always knock on the door of someone who bought recently and tell them that you are considering buying in the neighborhood and would like to their house.
Ask your agent to show you the comps that are active because there are comps and there are comps!!! I have seen real estate agents pull comps without having done their homework. They'll line up all the 3 BR 2 BA between 2200 and 2300 sf with a pool and they call them comps without even previewing them. A good agent will preview the comps and will tell you all there is to know about the properties. By the way, this is one of the reasons why real estate agents are so valuable. The good ones spend a huge amount of time visiting properties to know what's out there.
This being said, your game should be to purchase at the lowest price possible, isn't it? First thing you ought to do is find out the other player's motivation. You'd be foolish to propose $250,000 when you could get it for $240,000. Remember that if they don't accept your offer, you can always make another one.
Sellers should not get mad at you. You made an offer. If they want to get mad, tell them to get mad at the people who did not make an offer. Don't let the opponent what's in your hand. If you tell them that you absolutely love the place, they'll know that you will go way up. If you have another property that you are interested in and that is very similar, let them know. They will be softer.
If you don't like negotiating, then make your best offer and let your agent that it is final. But make sure that you stick to your guns. You do not want to lose credibility. In today's market, I have seen seller refusing offers only to come back two weeks later and asking the buyers to put the offer back on the table. This is why I would give them time to think about accpeting the offer. Don't give them only 24 hours to respond. Give them four days or more.
In your case, assuming that the comps are well chosen and there are no competing offers, I would bid below them. Good luck.
If your agent can produce comps that support lower numbers, go in with that lower number. A good buyers agent will be armed with market stats to support his/her buyers offer. By low, based upon the limited data here, I am thinking 231K to 235K.
Allow me to preface this response with a warning that I do not know your market or its conditions.
That said, if you were my client and if there were no other competing offers, I would suggest a starting offer at $237,500.
I would start by refining those "comps" to be sure they really are comps. Afterall, that is the number the appraiser from the bank will be using to justify your mortgage amount. If they both really are "comps" then you have an argument for an opening offer at the $238 range--just remember, you can insult the seller and they may not want to deal with you if you go in too low to begin with. Good luck!