A home I like has been sitting on the market much longer than everything else nearby. Does that give me more room to negotiate?

It can, but a long time on market does not automatically mean the seller will accept a large discount. The better question is why the home has not sold and whether the price, condition, presentation, disclosures, or previous deal history gives you a reasonable basis for negotiation.
Days on market are a clue, not the whole negotiation
A longer market time can point to pricing, condition, presentation, disclosure concerns, a previous escrow that did not close, or simply a seller who has not yet adjusted to current buyer feedback.
Build the offer around evidence
Look at price reductions, comparable sales, property condition, known repair needs, and the history that can reasonably be verified. That information can support a thoughtful offer more effectively than assuming a seller must accept a large discount just because the listing has been active longer.
Keep the strategy specific to the property
Dominique can help you review the listing history and nearby activity before deciding how aggressive or conservative to be. The goal is a defensible offer, not a promised discount.
California and San Fernando Valley context
The San Fernando Valley includes both fast-moving pockets and homes that need more negotiation or due diligence. Dominique can help you compare the real-estate side of the decision while inspectors and lenders address their licensed areas.
References
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