When the seller receives and accepts an offer, the listing agent will get escrow to send in an estimated HUD statement with the offer to the lender. That estimated HUD will also include all the outstanding liens including any back taxes and other liens like HOA.

The short sale lender will analyze the package and will respond to what they will pay for and what they will not pay for.

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I have seen them pay tax defaults. I have seen them balk on other defaults and ask the buyer to cure them.

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sdrealtor and others have seen cases where some lenders do keep property taxes current.