The vote will come two weeks before Democrats convene in Boston for their presidential nomination convention.
Several versions have been introduced, but there is wide support for one sponsored by Sen. Wayne Allard, R-Colo., which says, "Marriage in the United States shall consist only of the union of a man and a woman."
The bill then says states are not required to recognize a same-sex "marriage" approved by another state.
A similar version is being sponsored in the House by Rep. Marilyn Musgrave, R-Colo., though House leaders have not scheduled a vote.
Democratic lawmakers were mostly silent on the Republicans' decision to set the vote.
"I wasn't even aware that it had been" set, said Sen. Patrick Leahy, D-Vt.