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Yeah. Could use it to generate some urgency.
Worth noting that the viewer will know the place isn't currently 'officially' on the market so there won't be any competition in the wings, real or otherwise. They might offer you something low(er) because they think you might consider it or negotiate with them rather than going through the hassle of re-listing.
That said, if they don't offer you something you would consider you could always tell a massive lie in that there is another interested buyer who has made a better offer and maybe they would like to match it?
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I think in terms of what offer we'd accept there's a whole raft of competing things. Not having to remarket saves us hassle, but they're also benefitting from buying a house that isn't yet on the open market. Starting the process now might help get it finished before the stamp duty holiday ends in June, but that's actually slightly before we want to move. The lack of competition might get them thinking that they can make a stupidly low offer, but we can always say that the massive uncertainty in the market at the moment means that we want to give it the best chance on the open market unless they come up with a sensible offer.
Yup, when we terminated our contract with them they sent us a list of all the people who had viewed the house with a note that if any of them (or their relatives) bought the house in the future, then we'd still owe them the fees. That's pretty standard AFAIK. It's definitely something to raise with this buyer though i.e., make us an offer now because we won't let you buy it through a new agent unless your buying price covers double the fees.