The justices will consider whether to hear one or more of the cases at their first private conference of the new term, next Monday, and they may announce their choice or choices in the following weeks. If they do, they could hear arguments this winter and announce a decision by June.

The arguments for and against same-sex marriage are by now familiar to the justices, who considered but sidestepped them in a case from California last year.

Theodore B. Olson, a former United States solicitor general in the administration of George W. Bush, argued that case for the challengers of the California ban, and he is now one of the lawyers challenging Virginia’s ban. As before, he is joined by David Boies, his adversary in Bush v. Gore, the 2000 decision that delivered the presidency to Mr. Bush.

On the phone the other day, Mr. Olson listed the reasons to pick his case. It includes a class action, he said. It presents not only the issue of the right to marry but also that of whether states must recognize same-sex marriages performed elsewhere.

Virginia, he pointed out, was home not only to several of the giants who wrote the Constitution but also to Mildred and Richard Loving, who successfully challenged the state’s ban on interracial marriage in Loving v. Virginia in 1967.

“It’s pretty potent stuff,” he said of his case’s connection to another civil rights movement.

Mr. Olson was quick to add that the ultimate goal was victory, whatever the vehicle. “We have great respect for the lawyers in the other cases,” he said, “and we would be quite supportive of them if that’s what the justices want to do.”