Virginia would become the second state, after North Dakota, to add a "conscience clause" to state law under the bill before the General Assembly, The Washington Post reported.
The measure would allow state-funded, faith-based agencies to select which parents could adopt based on the agencies' beliefs.
The measure's backers say it would protect religious freedom by allowing birth parents to choose an adoption agency that follows its religious beliefs and, in turn, adoptive parents who share their beliefs.
"This measure will chisel into law the principle that people of faith can adhere to their convictions without fear of reprisal from those who would discriminate against their religious beliefs regarding how we should raise our children," said House Deputy Majority Leader C. Todd Gilbert, R-Shenandoah.
The House voted overwhelmingly to approve the bill last week, and the Republican-controlled Senate is expected to vote this week.
Republican Gov. Robert F. McDonnell is expected to sign the bill if it reaches his desk.
Critics, including the gay-rights group Equality Virginia and the ACLU, say the measure is discriminatory.
"Let's just speak the truth and tell it like it is,'' said Delegate David L. Englin, D-Alexandria. "This legislation is about ensuring that foster placement agencies that do not want to place children ... with same-sex couples are able to do that.''