It seemed like the softball question of the night when the Republican candidates for president were asked at Thursday’s debate to name a Hispanic politician they would appoint to their Cabinet.

Of course, the near universal answer was Sen. Marco Rubio, the popular freshman senator from Florida who is on virtually everyone’s short list of likely Republican presidential running mates. All eyes have been on the Cuban American tea party favorite, who has remained neutral in the primary but nevertheless has made his mark on the race.

Former Pennsylvania senator Rick Santorum was first to kiss the ring Tuesday, but front-runners Newt Gingrich and Mitt Romney were not far behind.

“I thought of Mr. Rubio in a slightly more dignified position,” Gingrich said, flashing a smile that suggested one would be right in guessing that he meant vice president.

For his part, Rubio has consistenly said he does not expect the nomination, nor would he accept it. But his popularity in this swing state, his widely lauded charisma, his credibility among tea party voters and his Hispanic roots make him an odds-on favorite for the bid.

“I believe in his heart at this time, his answer to a request to be a running mate would be no,” said Al Cardenas, former Florida GOP chairman and a mentor to Rubio. “But once somebody gets the nomination and that person who has that authority is the one calling you and talking to you, there’s a whole new patriotic mix to that question and you never know what the answer is going to be.”

Still, critics have raised questions about his suitability for the position. A year into his term, he has stumbled into some controversies, notably a Washington Post investigation that revealed he had misrepresented the story of his parents’ migration from Cuba to the United States.

Rubio, who used his compelling personal story in his insurgent 2010 campaign, characterized them as exiles who fled Castro’s rise. In fact, they were admitted to the United States as permanent residents 21 / 2 years before Castro took power. Rubio has said it does not change the basic thrust of their story, that they were alienated from their motherland because of the dictator.

Supporters say Rubio, a fluent Spanish speaker, could help the Republican Party with Hispanic voters, who are expected to cast critical swing votes in several states in the general election. Latinos have shifted heavily Democratic in recent years as the GOP took an increasingly harsh tone on the issue of illegal immigration.

But some Hispanics, including some conservatives, say Rubio has been a perpetrator of some of the most objectionable rhetoric as he rode a wave of tea party support to his Senate victory two years ago.

“I gotta say, some of our members on immigration specifically have expressed some fustration with what they believe to be his intransigence on this issue,” said Mario H. Lopez, president of the conservative-leaning Hispanic Leadership Fund.

Indeed, Rubio has taken a hard line on illegal immigration, a position that solidified his support among the tea party voters who rallied to his support. He opposes the DREAM Act, which would allow some children of illegal immigrants a path to citizenship, and spoke favorably of Arizona’s controversial anti-illegal immigrant measures.

Rubio also came out against the nomination of Supreme Court Justice Sonia Sotomayor, whose elevation to the high court was seen as a seminal moment by Hispanics across political lines.

In an indication of the difficulties he might face in a general election, a coalition of liberal, Hispanic pro-immigration groups protested his speech at the Hispanic Leadership Network conference Friday. Their message, spokesman Roberto Lovato said, was “Marco Rubio: Latino or Tea Partino?”

“We think it’s critical and urgent for the Latino community and the wider community to understand that Marco Rubio is a son of immigrants who’s anti-immigrant,” said Lovato, co-founder and strategist for the activist group “He’s a tea partier in Latino clothing.”