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.