WASHINGTON — Same-sex marriage advocates are eyeing the end of a decades-long fight as they ask the U.S. Supreme Court for a nationwide ruling that would cap a transformation in the legal rights of gay Americans.

The justices may say as soon as Friday whether they will hear arguments that the Constitution guarantees same-sex marriage rights. High court review would threaten bans that remain in effect in 14 states.

The country has already undergone sweeping changes since October, when the justices ducked the subject by rejecting seven appeals. The number of states where gays can marry has since surged to 36 from 19, with Florida’s addition this week. That puts more than 70 percent of the U.S. population in a gay-marriage jurisdiction and leaves a less daunting task for those justices inclined toward legalization.

“We are in the end game on the freedom to marry for same-sex couples,” said James Esseks, a lawyer who leads the gay-rights project at the American Civil Liberties Union. “The reality on the ground is that in 36 states same-sex couples can marry, and the sky has not fallen.”

At the same time, legal uncertainty has increased, putting more pressure on the Supreme Court to act. Federal appeals courts are now divided, with four backing marriage rights, one opposing and a sixth preparing to hear arguments in New Orleans on Friday.

The nine Supreme Court justices will have five appeals before them when they meet privately in their wood-paneled conference room on the court’s second floor. An immediate decision to take up one or more of those cases would probably mean arguments in April and a ruling by late June. The justices also could delay review until their next nine-month term starts in October.

A decision legalizing gay marriage would be a watershed moment for a movement that in 1996 had support from only 27 percent of the American public, a figure that now stands at 55 percent according to Gallup polls.