* Arizona Gov. Brewer says meeting was "cordial"
* Differences remain over border security, broad reform
* Did not address possible federal suit over state law
(Adds Obama comment, paragraph 5)
By Patricia Zengerle
WASHINGTON, June 3 (Reuters) - Arizona Governor Jan Brewer and President Barack Obama sat down for half an hour to discuss immigration on Thursday, but there were few signs that the two had bridged their sharp differences on the issue.
Brewer, a Republican, recently signed a controversial bill cracking down on illegal immigrants in Arizona that Obama has called "misguided" and over which his administration may file suit.
The law would require officers to determine the immigration status of any person they suspect of being in the country illegally.
Critics have said the measure, which takes effect on July 29, is a mandate for racial profiling.
"Although I understand the frustration of the people of Arizona when it comes to the inflow of illegal immigrants, I don't think this is the right way to do it," Obama said in an interview on CNN's Larry King Live after the meeting. "I think this puts American citizens, who... look Hispanic, are Hispanic, potentially in an unfair situation."
The law has added tension to relations with Mexico, whose president, Felipe Calderon, sharply criticized it during a visit to Washington last month, saying it could expose hardworking Mexican-Americans to discrimination.
Brewer said she and Obama had agreed to work together on immigration. But she insisted that border security should be a higher priority than the sweeping federal immigration reform favored by the Democratic president.
"I am encouraged that there's going to be much better dialogue," she told reporters after her meeting with Obama at the White House. Brewer said Obama told her he would send staff to Arizona within the next few weeks to address the issue.
Brewer said she and the president had discussed Obama's desire for comprehensive federal immigration reform, but said she felt border security should be the first priority.
"I believe the people of America want our borders secured, and after that we can tackle immigration," Brewer said. She said she wanted construction started on a border fence within the next month or so.
Arizona is the principal corridor for unauthorized immigrants entering the country from Mexico, and a busy entry point for Mexican cartels smuggling drugs into a voracious U.S. market.
As it sought Republican support for a sweeping immigration overhaul and sought to rally opposition to the Arizona law, the White House said last week it would seek $500 million for security and send up to 1,200 National Guard troops to the U.S. border with Mexico.
Brewer said she and Obama "brushed over" the possibility of a lawsuit in their "very, very cordial" meeting because Obama was going to leave that issue to the Department of Justice.
The White House said it had been a "good" meeting in which Obama pushed for overarching immigration reform and reiterated his concern about the Arizona law.
White House spokesman Robert Gibbs said Obama believes there should be a simultaneous focus on border security and broader reform. "The president believes that... those things have to happen together," he said at a daily news briefing. (Editing by Matt Spetalnick and Eric Walsh)