The Caleb Prouty House is tucked behind a tangle of bushes and brambles off Cromwell Lane, a vestige of 19th-century Greek revival architecture that survived the 1883 fire that destroyed much of downtown Vineyard Haven.

Stop & Shop acquired ownership of the old house in 2012, and as the company mulls redevelopment plans for a new Vineyard Haven store, it came before the Martha’s Vineyard Commission Thursday night with a somewhat probing request to demolish the home.

At a public hearing, Island attorney Geoghan Coogan said Stop & Shop wants to know what it can do with the vacant house before developing a broader plan for the Water street grocery store.

“We know that you are not going to allow us to demolish it. We are realistic about that,” Mr. Coogan said. “But we can’t develop the site with that house there. It doesn’t work for us. It’s too costly.”

Demolition of the house has come before the commission twice before, in 2013 and 2015. In the Public Archaeology Lab conducted a survey of the house and determined that it was eligible for listing in the National Register of Historic Places. Although it was never listed, the report said the house had historical significance.

“It stands as an excellent local example of a moderately high-style Greek Revival resident in Tisbury, remaining both a well-preserved exterior and almost pristine interior first floor,” the PAL report said.

Seven years later, the home is in a much more serious state of disrepair, Mr. Coogan said, showing the commission photographs of interior decay and rubble. He said the main wish is for the commission to allow Stop & Shop to relocate the home.

He said the store had looked into remodeling and using it as worker housing, but planners ultimately felt that wouldn’t efficiently serve the store’s housing needs.

“Our goal with this, just to be blunt, is to allow us to move it off the site,” Mr. Coogan said.

In a wide-ranging discussion, commissioners aired a variety of concerns. Some said a large part of the historic value of the home lies in its location, hidden away but facing the water, a rare vestige of historic downtown Vineyard Haven. Others thought a move might be a better way to acknowledge history in a more visible spot.

“For me, one of the big things that makes this a historic house is the location because it survived the great fire of 1883, being in the middle of the devastation,” commissioner Fred Hancock said. “The location is kind of important to its historic value. To judge whether it could be moved off site, I would like to see the rest of site plan.”

Mr. Coogan, countered by pointing out that most Islanders — and even most Vineyard Haven residents — are unaware of the property because it’s not visible. Commissioner Kathy Newman agreed.

“It’s in a place nobody notices,” she said. “It’s not contributing to history really. I guess it’s an interesting thing to think about. Sort of like the tree in the forest — if nobody hears it fall, there’s a little bit of that there. How can we make this little piece of history live again, because right now it is kind of dead?”

Commissioner Clarence (Trip) Barnes 3rd, who has moved houses, said this one probably wasn’t worth the effort. He said in his opinion the staircase was the only part of the house with true historic value. He estimated that more than 30 overhead wires would have to be taken out to accommodate a move too.

“Any funky fun old stuff has already been taken out of there,” Mr. Barnes said. “It’s had its day. Time to kiss it goodbye.”

Ben Hall Jr., an Island attorney representing abutting landowners, echoed the difficult predicament for both Stop & Shop and the commission.

“It’s really cool, but it’s such a tough move. There’s nowhere to take it without carving it up,” Mr. Hall said. “It has to go somewhere nearby but I don’t think there’s anywhere that would accept it . . . you guys have a tough decision.”

Commissioners noted that it would be easier to evaluate the merits of tearing down or moving the house if they had more perspective on the proposed Stop & Shop project itself.

Commissioner Gail Barmakian said she wanted to see a plan for the property if the commission allowed a move, and another plan it didn’t. Ben Robinson expressed similar sentiment.

“Stop & Shop could try to honor the history embedded in this site,” Mr. Robinson said. “Then if that doesn’t work, give concrete reasons why it wouldn’t work.”

The public hearing was continued to Nov. 7.