In an interview with BBC Jersey, external, Chief Minister Lyndon Farnham was asked whether his government would consider buying the site.
"I fancy that coming into public ownership somehow," he said.
Farnham said he was keen to speak to the National Trust about it.
"There's a great opportunity, if you look at the Simon Sand area along to Les Mielles, to create an area of outstanding natural beauty".
"That's something we can certainly take a look at" he said.
Farnham previously arranged for the purchase of Cafe Romany at Greve de Lecq following similar calls to protect the surrounding area.
With a guide price of £9.95m, Farnham said there were a number of possibilities that could be explored in any such purchase.
"There's public money if we decide to use it, there are other options as well with the National Trust and I've had a number of people in our 21E community, the wealthier residents, who are very keen to help contribute to that".
The National Trust of Jersey declined to comment at this stage.
