The case is one of "reverse discrimination." Twenty firefighters -- 19 white and one Hispanic -- who were denied promotions in New Haven, Conn., claim city officials discriminated against them because they were more concerned about potential complaints of Civil Rights Act violations than their performance on advancement exams.
The white firefighters say discrimination is discrimination no matter what color it takes, and therefore, the city did violate the Civil Rights Act in not promoting the white and one Hispanic firefighters.
How the justices rule could have a big impact on all businesses and governments that make job-related decisions involving race. In 2003, the high court said universities can consider race as one way to ensure student diversity.
But the case is also relevant because Sotomayor was one of the three appeals court judges who ruled for the city in the unanimous decision in the lower court.
A reversal by the Supreme Court that includes very critical remarks about the lower court ruling could be used as ammunition by some senators who don't want to see Sotomayor confirmed.
Sotomayor's views on race have been the focal point of criticism as she seeks a lifetime appointment to the highest court in the land. She has also been scrutinized for her statement outside the court that a "wise, Latina woman" would come to better conclusions more often than a white man.
Sotomayor's confirmation hearing is currently scheduled to begin on July 13. Senate Minority Leader Mitch McConnell told "FOX News Sunday" that her nomination must have a full airing before a vote, and that could mean delaying the hearing scheduled by Democratic senators, a scenario that is unlikely to happen.
"Just a day or so ago, we discovered that there are 300 boxes of additional material that has just been discovered from her time working with the Puerto Rican Legal Defense Fund," McConnell said. "The committee needs to have access to that material and time to work through it ... so we know all the facts before we vote on a person who's up for a lifetime job."
If confirmed, Sotomayor will replace Justice David Souter, whose retirement coincides with the end of the court's session on Monday. In April's oral argument of the firefighter case, Souter described it as a "damned if you do, damned if you don't" situation.
As Souter retires to New Hampshire, four justices are heading to Europe for summer teaching jobs, including in Austria, Ireland and Italy. But before they do, they also have to issue two other rulings and several orders on cases to accept for next session, which starts on the first Monday in October.
Among the rulings expected is whether campaign finance law can block movie producers from distributing politically-oriented films or networks from airing ads for those films. The case revolves around a movie distributed by Citizens United that blasted then-presidential candidate Hillary Clinton.
FOX News' Lee Ross and Caroline Shively contributed to this report.