Open question: Advice for offering on a house?

We are ready to make an offer on a house. Our absolute highest offer can only be closing $4,000 under the estimated price of the salesmen, plus costs. Are there any good strategies for offering smaller, without insulting it? Background on the possible motive of the salesman or lack of motive: In - they bought this house of a decade forwards, and even with our highest offer (smaller than, over for which they ask), they will obtain still another large profit. - You rent it out - It is only on the market for around one week, but we are the only prospective customer starting from nevertheless any proposals??