Carroll Rhodes, attorney for the group, says he believes the law will diminish minorities' voting strength, violating the 1965 Voting Rights Act.
Republican Gov. Phil Bryant has a May 18 deadline to sign House Bill 921, which would require voters to show a driver's license or other identification at the polls.
Because of Mississippi's history of racial discrimination, the state is required to get federal approval for any changes in election laws or procedures.
Secretary of State Delbert Hosemann says the voter ID law is closely patterned after Indiana's law that has been upheld by the U.S. Supreme Court. However, Rhodes says that doesn't matter because the histories of Mississippi and Indiana are different.