Baldwin told reporters Thursday she is “very confident” the bill will pass but said she needs “a little more time.”
Some Republicans, including Sen. Ron Johnson (R-Wisconsin), have signaled a need to hammer out legislative protections for religious liberty.
“There are some very legitimate concerns about religious liberty, and those concerns would have to be properly addressed,” Johnson said in an interview last week.
Johnson called the act “unnecessary” but said he saw “no reason to oppose it” in a statement in July.
A record number of 47 Republicans joined Democrats in passing the bill in the House in July.
The bill would repeal the 1996 Defense of Marriage Act, which defined marriage in federal law as the union of a man and a woman and permitted states not to recognize same-sex marriages that were contracted in other states.