Once you get the comps, figure out what you can afford--which you've already done, in this case--and how much the house is worth to you. Maybe the comps say it's worth $2.5 million, but for whatever reason you'd only pay $2 million for it. But if you really love it, maybe it's worth the full $2.5 million.
Then, you ask what does "eager to sell" mean. Generally, it means that there's some pressure on the owners to sell and they're willing to drop the price further. How much further? Again, who knows?
The fact that it's been ont he market for 2 months may not mean very much. In today's market, in many areas, the average time on market is over 2 months. In a blazing hot market, 2 months is a long time, but in today's cooler market, for higher priced homes, 2 months doesn't indicate much.
And you've got to understand the different reasons a property might be on the market for a long time. It probably means it's overpriced. Either that, or it doesn't show well at all. Yes, it can mean that the owner will be more flexible than when the property came on the market, and he/she was expecting a quick sale at close to list price. But, on the other hand, it can mean that the seller's inflexible. Maybe he/she has received offers more in line with comps, and he/she has rejected them as too low. It's always possible the seller will change his/her mind. But sometimes they're stubborn. ("No one's going to steal this house from me!")
Based on what you wrote, if the seller is "very eager," that's an invitation to come in low. I'd assume, from what you wrote, that if the 5% discount is "standard" (no, there really isn't a "standard"), that the 5% discount applies to houses properly priced, that show well, and whose owners would like to sell but aren't highly motivated. I'd suggest coming in at least 10% lower, probably more. If the owner accepts, good. If the owner counters, that's good, too.
Hope that helps.
It seems you're in the same boat we were/are in, found a house you like, not a dream home, but definitely something you would like to live in, but don't want to overpay. For us this was a large concern not to pay a 10+% premium. For our offer, we sent a long letter with reasons we priced the house as we did and why we believe our offer was fair. The reason for this is we also did not want to offend the sellers. The listing agent was very friendly but they declined to counter, c'est la vie. That said, the listing agent has basically given you carte blanche to come in with a number here. A "very eager" seller that has been described as such will be listening to all offers and should understand that they're being advertised in a way that will bring in lower offers. Personally, 2 million feels like a floor to me because its a nice round number that seller could have a mental barrier against. So if you came in at something like 2,050,000 with solid terms, I believe there is no reason for a "very eager" seller to be offended. You could also do something where you offer like 2.2 and have them pick up closing costs and put those towards pts on the mortgage to have a payment similar to paying 2 mil even.
To another point you made, I remember thinking after reading a previous post of yours with your price range and the fact you're looking WAAAAY up in westchester that you're probably a hot commodity right now. The areas up there, Bedford, Armonk, Pound Ridge, etc are beautiful with great schools and all, but they're not on train lines making commuting much more painful. With a range over 2 million and looking in those areas, I'd think there isn't a lot of competition on the buy side out there right now, but I'm not in that range nor a pro, so take it with a grain of salt.
Make an offer! What is the worst that can happen?
Don't worry about offending the seller. Seriously. If he offended by the offer, he isn't going to write a well reasoned letter to you explaining that you are offensive, or crash his grocery cart into yours at the store and scream to the world "YOU OFFEND ME!!!!"
I know that my analogy is inflatedly comical, but think about it. He gets offended for 2 seconds, tells his Realtor "no way will I go that low" and then he moves on. He won't even remember you name 1/2 an hour later. HOWEVER, what if he is that desperate and takes your offer? YOU WIN.
The other Realtors are right, make an offer and ride the wave. You might get lucky.