A rare portrait of John Barry, the man sometimes called the "father of the American Navy," recently went on the market, but it doesn't look like the valuable painting will be hanging at the Naval Academy.

The circa 1801 portrait by Gilbert Stuart has been held by Barry's descendants for most of its existence, but now it is being offered by an upstate New York art dealer for $300,000.

"We don't have an acquisition budget," said James Cheevers, art curator and associate director at the Naval Academy Museum. "If you would like to donate it, we'd be delighted to accept it; it is a pretty significant work."

Mr. Cheevers...