Both chambers have passed versions of the legislation over the last decade, but never during the same Congress. Some Republicans have raised doubts about whether the group behind the museum project could raise the estimated $500 million needed to build it. They also pointed to the dozens of museums devoted to women’s history throughout the United States.

Ms. Maloney, who first proposed the project in the late 1990s, said that other museums are devoted to women in particular areas of accomplishment — the arts, for example — but that there is not one that chronicles the overall contribution of women, in politics, civic life, war, science and other fields.

In fact, she said in an interview: “I don’t know of a national museum anywhere in any of the capitals of the world that chronicles the achievements of women. I find that astonishing.”

Some anti-abortion groups have also voiced concern that the museum would be a platform for advocating abortion rights.