Some Republicans were quick to challenge Ms. Napolitano’s claims that border security had significantly improved or that American workers would be helped by bringing illegal immigrants into the system.

“How can they claim that enforcement is done when there are more than 400 open miles of border with Mexico?” asked Representative Lamar Smith of Texas, the senior Republican on the House Judiciary Committee. He said the administration should “deport illegal immigrant workers so they don’t remain here to compete with citizen and legal immigrant job seekers.”

But Senator John Cornyn of Texas, the top Republican on the Judiciary subcommittee on immigration, agreed that it was time to open the immigration debate. “My commitment to immigration reform has not changed,” he said in a statement Friday. “I am interested in seeing a proposal sooner rather than later from President Obama.”

Senator Charles E. Schumer, Democrat of New York and the chairman of that subcommittee, has been writing an overhaul bill and consulting with Republicans, particularly Senator Lindsey Graham of South Carolina. Mr. Schumer said that the administration’s agenda was “ambitious,” but that he was “confident we can have a bipartisan immigration bill ready to go under whatever timeline the president thinks is best.”

Ms. Napolitano has been leading the administration’s efforts to gather ideas and support for the immigration overhaul, meeting in recent weeks with business leaders, religious groups, law enforcement officials and others to gauge their willingness to go forward with a debate in Congress.

Framing the administration’s proposals in stark law and order terms, she said immigration legislation should include tougher laws against migrant smugglers and more severe sanctions for employers who hire unauthorized workers.

Ms. Napolitano said that the Border Patrol had grown by 20,000 officers and that more than 600 miles of border fence had been finished, meeting security benchmarks set by Congress in 2007. She was echoing an argument adopted by Mr. Bush after the bill collapsed in 2007, and by Senator John McCain, Republican of Arizona, in his race against Mr. Obama. They said Americans wanted to see effective enforcement before they would agree to legal status for millions of illegal immigrants.

Some immigrant advocates were dismayed by Ms. Napolitano’s approach. Benjamin E. Johnson, executive director of the American Immigration Council, praised her package of proposals, but said some enforcement policies she outlined “have proven to do more harm than good.”