At the party's national convention in Tampa, Florida, a pair of cowboy-hatted Mexican-Americans from south Texas say they have the answer.
Be Sure To See If They Planned For This: RNC Delegate Offended By Presence of ‘Mexican’ At Disney’s Epcot Center
A walk through the halls at the Tampa Bay Times Forum confirms what poll after poll shows: the Republican Party of 2012 is a very, very white political movement.
Then, sauntering down the halls, come two rotund Mexican-American men. Both are wearing cowboy hats, with one in a sharp grey suit and the other a western-style shirt patterned after the lone star Texas flag.
Artemio Muniz and Art Martinez de Vara say they have a plan to bring conservative-minded Hispanic voters into the Republican fold. They have begun their campaign in Texas, and plan eventually to move on to California and the rest of the nation.
"The community is in play," said Mr Martinez de Vara, the Republican mayor of a heavily Hispanic Texas town outside San Antonio called Von Ormy.
Jeb Bush: 'Demographics are destiny'
"They're not an entrenched base for any party. No party can take it for granted."
But President Barack Obama, the Democratic candidate, took 67% of the Hispanic vote in 2008 and is poised again to win an overwhelming majority of that electorate this time around.
No less a senior Republican figure than Jeb Bush, the former governor of Florida and brother of former President George W Bush, has said he fears for the party's long term viability if it cannot broaden its base.
On ABC's Good Morning American on Thursday, he said Republicans were making a mistake by "sending a signal as it relates to issues to immigration, that it's only about border control, rather than about how we create an environment of economic growth".
"It's a gateway issue for the emerging voting population in our country," Mr Bush said. "Hispanic voters, Asian voters, the country's becoming increasingly diverse. My message was really about the long haul, you know, demographics are destiny, and if you just ignore it, you do it at your peril."Immigration concerns
Hispanic voters are hardly a monolithic bloc. In US demographic jargon, the term refers to anyone descended from a Spanish-speaking country. Polls indicate Hispanic voters are as concerned as non-Hispanic voters about the weak economy and the dismal job growth.
When we solve the immigration issue, it's over. If we can win that battle, really in a few years you're going to see the fruits of our labour”Artemio Muniz
But polls also show that they are far more concerned about immigration than non-Hispanic voters, and the Republican brand in recent years has taken on a hostile, anti-illegal immigrant tone that some interpret as anti-Hispanic.
President George W Bush attempted to pass immigration reform during his first and second terms, including guest-worker programmes and increased border security.
But in recent years Republicans have scuppered efforts to reform the nation's immigration system and to find a way to accommodate many of the more than 10 million illegal immigrants in the US, the vast majority of whom hail from Latin America.
Republicans in Arizona, Georgia and other states have pushed through laws prescribing harsh treatment of illegal immigrants that Hispanics fear encourage racial profiling.
All that has turned Hispanic voters, the fastest-growing segment of the electorate, off the Republican party, independent analysts and Republicans say.
At the Tampa convention, Mr Martinez de Vara and Mr Muniz say Hispanic voters are imminently reachable on many of the core Republican political positions - and say Democrats have become complacent.
The city of Baltimore is inviting immigrants to settle with no questions asked, BBC Mundo's William Marquez reports
"In Texas, we've been able to do the work," said Mr Muniz, chairman of the Federation of Hispanic Republicans. "We're creating a network, building a movement, winning races in Hispanic areas."
They pitch the core Republican message of muscular patriotism, fiscal responsibility and minimal taxes, coloured with a religious appeal to the devotedly Catholic Mexican-American voters.
"It's a very working class, hard working community," Mr Martinez de Vara of Von Ormy.
"The last thing they want is the government to come in and take the fruits of their labour, and the closer they are to Mexico generationally, the more they understand that."Building national support
When they encounter voters' concerns about the Republican party's anti-illegal immigrant posture, they turn the tables on Mr Obama. They note the record number of illegal immigrants deported during Mr Obama's tenure, as well as his failure to reform the immigration system.
They criticise Mr Obama's signature action on illegal immigration - a move this year to grant legal status to some young people who came to the US illegally as children. That move is temporary, they point out, and they stoke fears the young people will have to register their illegal parents with the government.
Interviews in Florida, a must-win swing state for both Mr Romney and Mr Obama, suggest that, immigration aside, Hispanic Americans are swayed by the same arguments, perceptions and prejudices as non-Hispanic voters.
|Cuban immigrant Elpidio Garcia: "Most of the Cubans I know, they vote Republican."|
Heidy Miner, an immigrant from Colombia and recently minted US citizen, said she would vote for Mr Romney. Above all, she said, she believed in his brand of aggressive capitalism, which she contrasted with President Barack Obama's "socialism".
"There is more opportunity for the middle class, the poor class, any class, to have an incentive or motivation to work for themselves and for the community and to move the country forward," she said, of an American governed by Mr Romney and the Republicans.
At a Cuban restaurant in Tampa, octogenarians Elpidio and Carmen Garcia, who fled Fidel Castro's Cuba decades ago, said their Christian religious beliefs and deeply held social conservatism drove them to back Mr Romney.
"He's against abortion and he's against marrying same-sex people," said Mr Garcia. "Most of the Cubans I know, they vote Republican."
Sitting on a stool nearby, Maria Salermo, also a Cuban immigrant, said she would back Mr Obama because he needed more time to carryout his economic recovery plans.
"He's doing things well," said Mrs Salermo, a 53-year-old caregiver to disabled children. "He's going to take more time to fix this place up. The Republicans are talking a lot of bullshit."
Having won over Hispanic voters in South Texas - and won elections - Mr Muniz and Mr Martinez de Vara say they are now working within the national party to build support for moderate immigration policies and rhetoric at the national level.
"When we solve the immigration issue, it's over," Mr Muniz said. "If we can win that battle, really in a few years you're going to see the fruits of our labour."