It would be a revival worthy of Lazarus, but President Obama is making a renewed push for an immigration overhaul, possibly during a lame-duck session of Congress after the November election — when members would no longer face an imminent political risk for supporting it.

Obama met with members of the Congressional Hispanic Caucus in the State Dining Room on Tuesday and discussed a strategy for passing a bill that had seemed dead for the year.

On Thursday morning, the president will put the issue before the American public. In a speech at American University, he plans to make the case for providing a path to legal status for the estimated 11 million people who live in the U.S. illegally while strengthening border enforcement.

White House Press Secretary Robert Gibbs said at his daily briefing Tuesday that “this continues to be a very important national issue” requiring Republican support. To date, no Republican senators have agreed to back a comprehensive immigration bill. Nor has such a bill been introduced in the Senate.

Obama “can’t sign something that doesn’t exist,” said one person who was at the White House meeting.

As recently as May, Obama said he merely wanted to “begin work” on immigration this year — not complete a bill. But this week he has approached the issue with renewed urgency.

He spoke to immigration advocates at the White House on Monday, setting aside time from coping with the oil spill in the Gulf of Mexico and a shakeup in his military command in Afghanistan.

Latino lawmakers who have criticized the White House for neglecting immigration said they were pleased.

Rep. Luis V. Gutierrez (D-Ill.), who was part of the Hispanic Caucus meeting with Obama, said in an interview: “He’s going to speak to the nation on Thursday and tell the country why it’s important to have comprehensive immigration reform. That’s something we’ve been demanding of this administration.”

But advocates have heard assurances before.

Deepak Bhargava of the Washington-based Center for Community Change was among those who met with the president Monday. In an interview afterward, Bhargava said Obama “was unambiguous about his commitment. The question is whether the actions will match the words over the next few weeks.”

In their hourlong meeting Tuesday, lawmakers and the president debated a strategy for passing a bill in the coming months — no small task given that members are increasingly focused on the upcoming election, and national polls show broad support for Arizona’s strict new immigration law.

With conservatives energized, angry and likely to storm to the polls, Democrats fear they will lose even more seats in Congress than a president’s party typically does at the halfway point in his term.

Voting on an immigration bill in a lame-duck session has some advantages in proponents’ eyes. Outgoing members of Congress would have little reason to fear backing a controversial bill. And those who won might be more likely to support it, since they wouldn’t have to face voters for another two years — when Obama is up for reelection and likely to draw progressives to the polls.

In addition, if Republicans make major gains in November, an immigration overhaul could be impossible in 2011 or 2012.

While running for president, Obama pledged to act on immigration in 2009. That deadline came and went. But Obama won two-thirds of the Latino vote in 2008 and has no wish to alienate a growing constituency.

Raising the issue anew allows Obama to mollify his Latino supporters. But it also puts Republicans in a tough spot. Neither party can afford to write off a Latino community whose influence is growing.

Forcing a vote on immigration would give Republicans a difficult choice: They could vote against the bill and risk antagonizing Latinos, or vote yes and invite the wrath of “tea party” activists and other conservatives opposed to what they view as amnesty for illegal immigrants.

In his private meetings this week, Obama has emphasized that Republicans are the main force blocking a bill.

“He said over and over again the Republican obstruction was the key to preventing this from getting done,” said Eliseo Medina, international executive vice president of the Service Employees International Union, who took part in a meeting with Obama on Monday.

In another move likely to please Latino voters, Obama’s immigration enforcement chief, John Morton, issued a memo Tuesday ordering his agency to focus on deporting criminals and those who pose a national security threat, rather than on pursuing people such as “immediate family members of U.S. citizens” and those caring for children.

Morton has long embraced those priorities publicly. The memo was an effort to make them clear to every employee of Immigration and Customs Enforcement, said a senior ICE official who was not authorized to speak publicly.

The Los Angeles Times has reported in recent months on deportation cases against college students and others with no criminal records, including one against a Nevada couple that a federal judge criticized as “horrific.” After the article appeared, immigration officials told the family it would not be deported.

Officials in the Department of Homeland Security, which encompasses ICE, say it is difficult for senior officials to learn of every such case wending its way through the sprawling bureaucracy.

The Morton memo orders immigration officials to focus on removing “aliens engaged in or suspected of terrorism or espionage, or who otherwise pose a danger to national security; aliens convicted of crimes, with a particular emphasis on violent criminals, felons and repeat offenders; aliens not younger than 16 years of age who participated in organized criminal gangs; aliens subject to outstanding criminal warrants; and aliens who otherwise pose a serious risk to public safety.”

peter.nicholas@latimes.com

Ken Dilanian in the Washington bureau contributed to this report.