Aha, the helicopter vultures ah gathering! Wanting to feast on your mum’s estate before the poor soul is even dead…
OK, first off, get an estate agent (Do NOT mention the relative keen to buy!) to come and do an informal valuation of the house. They should do this for free. They should give you a price that is ‘this is what we will put it on the market for’, and one that is a ‘this is what I think you will achieve’.
Ask the agent how long it will take to sell. Brexit is really screwing up house sales at the moment!
Also, check on the property websites (Right move, etc etc) for what similar properties to your mum are going at.
Finally, if you want, you could pay a surveyor to give you as accurate a valuation as he possibly can. (Remmber, this is what purchasers do when they have a survey done - part of what the surveyor does is to value the property to see if you are overpaying etc).
This will give you a more realistic idea of the price your mum is likely to get from the sale. You can compare with the vulture’s offer.
You should also factor in the COST of selling through an estate agent - they usually charge 1.5% of the achieved price, but this may vary. If you sell ‘privately’ this cost won’t be there. (However, of course, there will still be the cost of conveyancing as the seller) (do NOT sell without a lawyer overseeing the transaction - not when you are dealing with a vulture!!!)
However, as well as price the second factor is time. If your mum needs to sell up fast, to free up funds for the care home fees, then that has to be taken into consideration. Selling through an estate agent is highly uncertain until the contract is signed (see Brexit effect above).
All that said, in principle, if the vulture wants to buy your mum’s property, they can damn well do so at the market rate. Why let them ‘steal’ from your mum?
Don’t be bullied or harassed in this! As you say, you have a LEGAL ‘duty of care’ as PoA to ensure your mum’s interests are paramount, or you could open yourself to charges of ‘embezzlement’ or whatever. Be cautious!