McBride will be the first transgender member of Congress when she takes office in January.
South Carolina Republican Representative Nancy Mace recently introduced the measure.
If passed, House members, officers, and employees would not be allowed to use restrooms designated for the opposite biological sex.
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"The sanctity of protecting women and standing up against the Left's systematic erasure of biological women starts here in the nation's Capitol," Mace wrote in a statement.
McBride later posted on X, stating in part, "This is a blatant attempt from far right-wing extremists to distract from the fact that they have no real solutions to what Americans are facing."
She noted the examples of housing, health care, and child care as issues more important than "culture wars."
Mace's statement, and the bill that is expected to follow, is the latest chapter in the long fight over women's restrooms in the Capitol.
The Women's Reading Room in Stat Hall is now named for former Rep. Lindy Boggs (the mother of ABC legend Cokie Roberts) and was a longtime private area for women.
And in 2011, Speaker John Boehner directed the Architect of the Capitol to build a women's room off the House floor - in an old office that used to be for the Parliamentarian.
If Mace's measure were passed, its management would fall under the jurisdiction of the sergeant at arms, which is the chief law enforcement officer in the House of Representatives.
ABC News contributed to this report.