The conservative justices on the U.S. Supreme Court Wednesday sounded very skeptical of the Obama administration’s argument that Arizona’s 2010 immigration law is a major overreach by the state into federal power; now they could deal two huge blows to President Obama’s policy agenda in June.

In filing suit against one of the most stringent immigration measures in the country, the Obama administration has argued the most controversial element of the provision: a requirement that police officers check the immigration status of people they suspect are in the country illegally constitutes the state taking the power of enforcing immigration laws into its own hands, instead of ceding it to the federal government.

But many of the justices on the Court’s conservative bloc argued Arizona was simply enforcing existing federal law on immigration with its provision. The law, those justices noted, simply requires Arizona officials to notify the federal government someone is in the country illegally; it doesn’t mandate arrests or other punishments, such as deportation.

“It seems to me the federal government just doesn’t want to know who is here illegally,” said Chief Justice John Roberts.

At one point, even Justice Sonia Sotomayor, an Obama appointee who is the Court’s first and only Hispanic member, told the U.S. government attorney, Solicitor General Donald B. Verrilli, his position was not “selling very well” with the justices.

One of the most polarizing issues about the law, the idea that it could lead to increased suspicion of any Latino person in Arizona, was not actually discussed at the hearing. The U.S. government has not attacked the law on the grounds that it constitutes racial profiling, even as outside groups, such as the American Civil Liberties Union, have made that case.

“No part of your argument has to do with racial or ethnic profiling?” Roberts asked Verrilli, who agreed.

Instead, the case turned into a debate on the power of the federal government and states that resembled last month’s arguments on health care reform. The conservative justices largely attacked the Obama administration’s view of federal power in that case as well.

On Wednesday, Verilli argued that, for both domestic and foreign policy reasons, the federal government should have “exclusive” authority on immigration. He warned of “mass incarceration” of people the Arizona government suspected did not live legally live in the United States.

The justices seemed far from convinced. Now, when the court announces many of its rulings in June, there is the possibility the justices could strike down Obama’s signature health care law and then uphold the Arizona immigration measure, which the president has strongly opposed.

Follow Perry Bacon Jr. on Twitter at @perrybaconjr