The case was brought to the Supreme Court by The Slants after their trademark was refused because the name is disparaging to Asian-Americans.
The Asian-American band acknowledges the name may offend, but say they chose it to "reappropriate" its meaning.
The ruling could impact the high-profile case of the American football team, the Washington Redskins.
In 2014 the US Patent and Trademark Office cancelled the team's six trademarks, including the lucrative team logo, after years of complaints from Native American groups.
The Slants had first attempted to register their band name in 2011, and sued after they were refused.
The Portland, Oregon "dance-rock" band, which is made up entirely of Asian-Americans, claimed that they chose the name to take back the racial epithet.
A lawyer for bandleader Simon Tam argued that his client "was following in the long tradition of 'reappropriation,' in which members of minority groups have reclaimed terms that were once directed at them as insults and turned them outward as badges of pride."
On Thursday, the US Supreme Court agreed to hear the band's case. Hearings will begin next week and are expected to last several months.
Redskins owner Dan Snyder has said that the team's name honours Native-American groups, and has long refused to change the name.
President Obama has repeatedly suggested that it is time for the team name of the US capitol city to change.
The nation needs to "break stereotypes", Mr Obama said in 2015 at a White House Tribal Nations conference.
"I believe that includes our sports teams," he added.
Redskins lawyers note that other offensive names have previously been approved the patent office, including trademarks for Yellowman, Red Man, Black Tail, Jap and Moonie.
The court is not expected to issue its ruling for several months.
(The Slants claim they are "reappropriating" the offensive term)