The case has split the Bush administration and could have far-reaching ramifications for the nation's racial landscape.
The court heard two hours of arguments on the extent to which race can be a factor in admissions to colleges and universities.
The case was brought by three white students who challenged the University of Michigan's policy of considering race as one factor among many in the way it selects its students.
The nine-member court has been closely divided between conservative and liberal factions. Much of the attention during arguments will be on Justice Sandra Day O'Connor, a key swing vote, who could control the outcome. Yesterday she questioned a lawyer who argued that race cannot be considered at all.
"We have given recognition to race in a variety of settings," she said. But she expressed concern that the university's policy continued indefinitely while other programmes the court has supported operated for a fixed time period.
Michigan and its law school give extra credit to ethnic minority applicants, as well as to applicants from low income backgrounds and to those from underrepresented states.
The university argues that a diverse student body is a goal that benefits all students. The plaintiffs argue they were denied a place at the university because they are white.
"I was treated unfairly because of my skin colour," said Jennifer Gratz. "Court records show that if I had been black, Hispanic or Native American, I would have had a nearly 100% chance of admission with my grades and record."
The issue has provoked public splits in the Bush administration. In January the White House filed papers with the supreme court urging it to find against the university.
Mr Bush also spoke out against college admissions policies that "unfairly reward or penalise prospective students based solely on their race".
The national security adviser, Condoleezza Rice, backed the president, while insisting that she believes race can and should be a factor in admissions policies. She acknowledged that she had been a beneficiary of such a system at Stanford.
Meanwhile, the secretary of state, Colin Powell, has openly disagreed: "Whereas I have expressed my support for the policies used by the University of Michigan, the president, in looking at it, came to the conclusion that it was constitutionally flawed, based on legal advice he received."
Yesterday Justice John Paul Stevens asked if the programme might generate racial hostility. Maureen Mahoney, a lawyer for the university, replied that the policy sought to minimise resentment and said there was overwhelming support for it among its students.
Justice Antonin Scalia, a staunch opponent of affirmative action, appeared most hostile. "Some applicants are given a preference because of their race," he said. "Once you use the term 'critical mass', you are into quota land."
The last time the court visited the issue was 25 years ago, in University of California v Bakke. That ruling struck down quota systems but left some room for race to be a factor in university admissions.