By Lawrence Hurley

WASHINGTON (Reuters) - The U.S. Supreme Court on Friday scheduled oral arguments in a major dispute on transgender rights for March 28, when the U.S. Senate is set to be in the midst of a political fight over President Donald Trump's nominee to a vacant seat on the bench.

By March, the Republican-controlled U.S. Senate is likely to be deliberating on whether to approve Neil Gorsuch, a conservative federal appeals court judge from Colorado, to the court. Where Gorsuch stands on social issues like transgender rights is likely to be a much-discussed question during the confirmation process.

Republicans are hoping the Senate Judiciary Committee holds hearings and votes on the nomination by late March, paving the way for a vote in the full Senate the first week of April, before the chamber begins a two week spring recess, according to Senate aides.

In the transgender case, in which the eight-justice court could be split 4-4 without a ninth vote, a Virginia public school district is fighting to prevent a female-born transgender high school student from using the boys' bathroom. The dispute involves a transgender student named Gavin Grimm, who identifies as male and sued in 2015 to win the right to use the school's boys' bathroom.

At the heart of the case is the question of whether transgender people are covered by a ban on gender discrimination in education under federal law. The administration of former President Barack Obama said it was. The Trump administration has not yet weighed in.

Until the Senate approves a nominee, the court remains one justice short following the February death of Antonin Scalia, which left it with four conservatives and four liberals. That raises the possibility of a 4-4 ruling that would leave in place the decision favoring Grimm by the Richmond-based 4th U.S. Circuit Court of Appeals. A 4-4 ruling would set no nationwide legal precedent. It is also possible that the court could rehear the case if Gorsuch is confirmed.

(Reporting by Lawrence Hurley; Editing by Tom Brown)