Despite steep odds, the White House has discussed prospects for reviving a major overhaul of the nation’s immigration laws, a commitment that President Obama has postponed once already.

Obama took up the issue privately with his staff Monday in a bid to advance a bill through Congress before lawmakers become too distracted by approaching midterm elections.

In the session, Obama and members of his Domestic Policy Council outlined ways to resuscitate the effort in a White House meeting with two senators -- Democrat Charles E. Schumer of New York and Republican Lindsey Graham of South Carolina -- who have spent months trying to craft a bill.

According to a person familiar with the meeting, the White House may ask Schumer and Graham to at least produce a blueprint that could be turned into legislative language.

The basis of a bill would include a path toward citizenship for the 10.8 million people living in the U.S. illegally. Citizenship would not be granted lightly, the White House said. Undocumented workers would need to register, pay taxes and pay a penalty for violating the law. Failure to comply might result in deportation.

Nick Shapiro, a White House spokesman, said the president’s support for an immigration bill, which would also include improved border security, was “unwavering.”

Participants in the White House gathering also pointed to an immigration rally set for March 21 in Washington as a way to spotlight the issue and build needed momentum.

Though proponents of an immigration overhaul were pleased that the White House wasn’t abandoning the effort, they also wanted Obama to take on a more assertive role, rather than leave it to Congress to work out a compromise.

Immigration is a delicate issue for the White House. After promising to revamp in his first year of office what many see as a fractured system, Obama risks angering a growing, politically potent Latino constituency if he defers the goal until 2011.

But with the healthcare debate still unresolved, Democrats are wary of plunging into another polarizing issue.

“Right now we have a little problem with the ‘Chicken Little’ mentality: The sky is falling and consequently we can’t do anything,” Rep. Raul M. Grijalva (D-Ariz.) said in an interview.

Republicans are unlikely to cooperate. On Capitol Hill, Republicans said that partisan tensions had only gotten worse since Obama signaled this week that he would push forward with a healthcare bill, whether he could get GOP votes or not.

Sen. Jeff Sessions (R-Ala.) said in an interview, “The things you hear from the administration won’t be well received.”

Schumer, speaking as he walked quickly through the Capitol, said he was having trouble rounding up Republican supporters apart from Graham. “It’s tough finding someone, but we’re trying,” Schumer said.

On Thursday, Schumer met with Homeland Security Secretary Janet Napolitano, who oversees the government’s immigration efforts, to strategize over potential Republican co-sponsors.

“We’re very hopeful we can get a bill done. We have all the pieces in place. We just need a second Republican,” Schumer said in a statement.

Among proponents, there is a consensus that a proposal must move by April or early May to have a realistic chance of passing this year. If that deadline slips, Congress’ focus is likely to shift to the November elections, making it impossible to take up major legislation.

“There’s no question that this is a heavy lift and the window is narrowing,” said Janet Murguia, president and chief executive of the National Council of La Raza, a Latino advocacy group.

When it comes to immigration, Obama’s strategy echoes that of healthcare. He has deferred heavily to Congress, leaving it up to Schumer and Graham to reach a breakthrough with the idea that he would put his weight behind the resulting compromise.

peter.nicholas@

latimes.com