Republican senators are inching toward working on a broad immigration overhaul with their Democratic colleagues, a sign that the window of opportunity to address the issue could again be cracking open, at least in the Senate.

The opening comes as President Barack Obama has been making increasingly forceful statements about his desire for Congress to pass a sweeping immigration bill.

“My expectation is that we get a bill introduced and we begin the process in Congress very soon after my inauguration,” Obama said at his press conference Wednesday.

“Before the election … I predicted that the Latino vote was going to be strong, and that that would cause some reflection on the part of Republicans about their position on immigration reform,” the president said. “I think we’re starting to see that already. We need to seize the moment.”

Almost three-quarters of Hispanic voters supported Obama’s re-election, with a higher percentage backing the president in some swing states such as Colorado. Republicans in Congress have subsequently started talking about the need for an immigration overhaul.

“I think most of them are convinced,” said Sen. John McCain, R-Ariz., who led the last major congressional push to overhaul immigration, which collapsed in 2007. “We’ve got to get the issue resolved.”

Sen. Orrin G. Hatch, R-Utah, also said Wednesday that he thinks the time is right to bring up immigration again.

“I’ve always been empathetic toward them solving this problem,” Hatch said.

Democrats, delighted by the Republican change in tone, are vowing to make an aggressive push on the issue, which has traditionally been one of the most intractable in Congress.

Already, Sen. Lindsey Graham, R-S.C., has been meeting with Sen. Charles E. Schumer, D-N.Y., the third highest-ranking Democrat in the chamber, to figure out possible areas of compromise.

Sen. Marco Rubio, R-Fla., said he would revive his effort to craft a proposal to give legal status to young people brought to the United States illegally as children and who know no other home. Democrats have backed the DREAM Act (HR 1842, S 952) to give such young people a direct path to citizenship, but Rubio’s version would not go that far, in order to pick up Republican support.

And Sen. Susan Collins, R-Maine, said this week that she could support a version of the DREAM Act, even if it included a path to citizenship.

“I believe that we should start with the DREAM Act bill,” Collins said Tuesday. “Children who are brought here through no fault of their own clearly should be treated differently from an adult who sneaked in across the border.”

Collins said she would have supported the bill the last time it came up on the Senate floor in 2010 if provisions requiring that eligible young people stay out of legal trouble had been stronger.

Obama reiterated his commitment to the DREAM Act during his press conference.

“The one thing I’m very clear about is that young people who are brought here through no fault of their own, who have gone to school here, pledged allegiance to our flag, want to serve in our military, want to go to school and contribute to our society, that they shouldn’t be under the cloud of deportation,” he said. “That we should give them every opportunity to earn their citizenship.”

But the willingness of a growing number of Senate Republicans to address long-standing immigration problems does not make it easier to reach a consensus on Capitol Hill. A critical point of contention remains whether to grant a path to citizenship to all of the 11 million people who live in the United States illegally.

Many Democrats want granting citizenship to be part of a comprehensive overhaul.

Republicans were less willing this week to stake out specific policy positions but said that anything that would grant citizenship to undocumented people would be a difficult sell in the party.

Sen. Saxby Chambliss, R-Ga., suggested Wednesday that he would not support granting citizenship. He hedged when asked whether he would back legalizing the status of the undocumented without necessarily granting them citizenship.

Even if the Senate is able to hash out a compromise, there is no guarantee the House will endorse it.

House Speaker John A. Boehner, R-Ohio, said last week that he would favor a “step by step” approach to overhauling immigration laws, an indication that he is not willing to go along with the Senate Democrats’ strategy of introducing a single sweeping piece of legislation. And it remains to be seen whether conservatives in Boehner’s caucus will go along with his approach.

Still, despite the difficulties that are sure to arise in the months ahead as Congress begins work on an immigration overhaul, Senate Democrats are refusing to retreat from their postelection optimism.

“No one expects John Boehner to do a 180-degree about face in 24 or 48 hours, so the fact that he seems to be moving a little bit in the right direction is a good thing, and I’m positive about it,” Schumer said.

Niels Lesniewski and Emily Ethridge contributed to this report.