The financial picture has grown so bleak at the American Folk Art Museum that its trustees are considering whether to shut it down and donate its collections to another institution, said a person involved in the discussions, who requested anonymity because the talks are confidential.

No final decision has been made, and members of the folk museum’s staff are said to be lobbying to keep it going in some form. But the museum’s leadership has been in talks with the Smithsonian Institution for several months about possibly acquiring the collection in conjunction with the Brooklyn Museum.

A decision to dissolve the museum and transfer its collection would require the approval of both the New York State attorney general’s office and the State Department of Education. The attorney general’s office would consider, among other things, whether the transfer would put New York State residents at a disadvantage.

Even if ownership of the collection were transferred to the Smithsonian, one possibility being discussed is to have the Brooklyn Museum display some of it long term, making it still accessible to New Yorkers. The folk art museum has one of the country’s finest collections of American folk art, including some 5,000 quilts, paintings and functional objects like weathervanes. But it has long been plagued by serious financial problems.

A decade ago the museum borrowed $32 million, in the form of bonds, to finance the construction of an impressive building on West 53rd Street in Manhattan, designed by the architects Tod Williams and Billie Tsien. In 2009, after suffering substantial investment losses in the financial crisis, the museum defaulted on its debt, and in May the trustees decided to sell the building to the Museum of Modern Art, down the block, in order to pay off the debt.

But the sale of the building for $31.2million, while covering the debt, did not leave the museum with any extra cash. And it left, if anything, an even more difficult conundrum for the museum about how to move forward.

The museum still has a small, 5,000-square-foot space in Lincoln Square, at Columbus Avenue near 66th Street, where it could continue mounting small shows if the trustees felt that it could sustain the costs of keeping a staff and paying to store, conserve and insure the collection. The museum remains open at that location these days, and admission is free. The building on West 53rd Street is closed.

The museum’s president, Laura Parsons, said of the current situation: “The board took the first step of discharging its obligations to the bondholders — the next step is to determine what the best outcome for the museum and the art is.”

Ms. Parsons declined to go into greater detail, saying that the discussions were confidential.