Republican leaders and strategists have wracked their brains since President Barack Obama’s re-election to reckon how the party can broaden its appeal to minorities.

They have yet to figure it out, but they’re working on it — and furiously. Some of the same conversations took place after the 2008 elections, but as Obama is inaugurated for a second term and the party’s base in presidential election cycles continues to shrink as a proportion of the electorate, it’s become abundantly clear to GOP leaders that a new winning formula is necessary.

“When you do the math, it’s a very narrow path to victory in a presidential year without significant improvement among Hispanics,” Republican pollster Glen Bolger said.

In recent days, Republican National Committee Chairman Reince Priebus attended a minority outreach summit in Texas and former Florida Gov. Jeb Bush led a Hispanic Leadership Network board meeting in the Sunshine State. Their goal: expand the GOP’s base and potential for swing votes, especially among Hispanics, the fastest-growing segment of the electorate.

Sen. Marco Rubio, a potential 2016 presidential candidate, has in the past week begun a push for sweeping immigration overhaul legislation. The Florida Republican laid out the key points of his plan in an interview with The Wall Street Journal editorial board, calling immigration a “gateway issue” for Hispanics.

In interviews with GOP strategists and Capitol Hill sources, there was a consensus that vastly improving the party’s relationship with Hispanics — and minorities as a whole — likely entails a combination of an immigration overhaul and an alteration of the rhetorical tone Republicans use when discussing the issue.

“There are conversations going on all over town and all over the country among Republicans about how to address a changing electorate,” said GOP pollster Whit Ayres, who conducted a postelection poll of Hispanics in four states that found the GOP brand toxic. “It’s not just Hispanics, but Hispanics are obviously the largest and fastest-growing group, particularly in critical swing states.”

Terry Nelson, who served as national political director for President George W. Bush in 2004, when the Republican won 44 percent of the Hispanic vote, said there are certainly mixed feelings within the party on what must be done on immigration, but the dialogue over the past several years has clearly alienated Hispanic voters.

“I think that the Republican Party and political leaders have to think about how we have a constructive immigration policy for the millions of people who are here illegally, how we have a constructive immigration policy for those who want to come here and work, and how we talk about it in a more constructive way,” Nelson said.

The reasoning is straightforward: The influence of white voters as a percentage of the electorate continues to trend downward. It reached an all-time low of 72 percent last year, when Hispanics accounted for 10 percent of the electorate for the first time. GOP presidential nominee Mitt Romney won white voters by an astounding 20 points but lost the election by 126 electoral votes.

Romney won just 27 percent of Hispanic voters, the lowest percentage for a Republican in a two-way presidential race since 1976. Ross Perot’s presence on the ballot in 1992 and 1996 as a third-party candidate pushed the GOP’s take among Hispanics to a quarter of the vote or less.

“This is incremental,” Bolger said. “We don’t go from 30 [percent] to 50 percent overnight. It’s a long, slow slog, but it’s something that has to happen.”

The party has a few Hispanic statewide elected officials that it hopes can continue to serve as an entrée to the community: Rubio, Texas Sen. Ted Cruz, and Govs. Susana Martinez of New Mexico and Brian Sandoval of Nevada — the latter two from swing states that have tilted toward Democrats in the past two elections.

The problem, GOP sources said, is that the loudest Republican voices on immigration tend to emanate from the party’s hard-liners. That can be an issue in presidential primaries, as candidates often try to outflank each other on the right.

“There’s a widespread consensus that we have got to change the tone by which we talk about undocumented immigrants, immigrants in general and Hispanics in particular,” Ayres said. “I don’t know that there’s any consensus yet on the policy changes that need to occur, but that is part of the conversation.”

Rubio, for one, appears to be headed down that road with an immigration overhaul plan, the heart of which was unveiled in recent media interviews. Rubio, whose candidacy for president alone could give Republicans an inroad among Hispanics, has been meeting in recent months with members of both parties in the House and Senate, as well as outside groups active on immigration.

“I’ve disagreed with some ideas offered in past debates and the way the issue’s been handled,” Rubio said in a statement to CQ Roll Call. “So it’s our responsibility to offer solutions that modernize our legal immigration system, strengthen security and enforcement measures, and deal with the undocumented population in a humane way that doesn’t give them a special advantage over immigrants trying to come legally.”

There is pressure on both parties to do something on immigration in the current Congress. Senate Majority Leader Harry Reid, D-Nev., called it his top priority in an interview with Las Vegas’ Public Broadcasting Service station. But moving immigration legislation has proved a tall task, as evidenced most recently by the 2007 congressional battle that came up empty.

Republicans argue that Democrats have done a good job co-opting the messaging on immigration, making Republicans seem unwilling to do anything on the issue.

“The problem that you always ran into is that the price of admission on the things that 60 percent of people agreed on were things that virtually nobody agreed on,” a Senate GOP aide said. “So your blanket amnesty discussion was always the price of admission for what you do with HB1 visas, or what you do with guest workers.”

Rubio’s proposal, which has yet to be finalized, is largely at the mercy of Reid, who controls what legislation comes up for a vote on the floor of the Senate, and what will likely be a divided GOP, which could fracture along the line of members concerned about primary challenges. A spokesman in Reid’s office did not respond to an email seeking comment, but Rubio’s public push for legislation came up at a White House press briefing last week.

“The reports about Sen. Rubio’s ideas bode well for a productive bipartisan debate, which we hope will start in earnest soon after the inauguration,” White House Press Secretary Jay Carney said Tuesday. “We hope that it signals a change in the Republican approach to this issue, because if we are going to get this done it’s going to take more than just a handful of Republicans working across the aisle.”

For the GOP, there is no simple prescription, but Bolger said having strong Hispanic candidates running for office, changing the way the party talks about immigration and coming up with sensible legislation are key to winning support in elections.

“That’s not going to be easy,” Bolger said, “but we can’t keep giving them the finger 18 months out of an election cycle and then expect the last six months to play footsie with them.”