He added, “It is such a toxic subject, such an important time for Democrats.”

The administration seemed to be taking a carrot-and-stick approach on Sunday. Homeland Security Secretary Janet Napolitano, in town to give the governors a classified national security briefing, met one-on-one with Jan Brewer, the Republican who succeeded her as governor of Arizona and ardently supports the immigration law.

About the same time as that meeting, Attorney General Eric H. Holder Jr. said on a taped Sunday talk show that the Justice Department could bring yet another lawsuit against Arizona if there is evidence that the immigration law leads to racial profiling.

Ms. Brewer said she and Ms. Napolitano did not discuss the current lawsuit. Instead, in a conversation she described as cordial, they discussed Arizona’s request for more National Guard troops along the border with Mexico, as well as other resources.

The Democrats’ meeting provided a window on tensions between the White House and states over the suit, which the Justice Department filed last week in federal court in Phoenix. Nineteen Democratic governors are either leaving office or seeking re-election this year, and Republicans see those seats as crucial to swaying the 2012 presidential race.

The Arizona law — which Ms. Brewer signed in April and which, barring an injunction, takes effect July 29 — makes it a state crime to be an illegal immigrant there. It also requires police officers to determine the immigration status of people they stop for other offenses if there is a “reasonable suspicion” that they might be illegal immigrants.