Republican leaders have begun reckoning with the fact that their party has grown increasingly out of step with a broad majority of American voters.

While party leaders remain confident in their beliefs, they have identified a litany of problems and a steep set of challenges: flawed candidates, a problematic message, the alienation of nonwhite Americans who account for a growing share of the population, outdated technology and a political operation that is not up to that of the Democrats.

A telling sign of their determination to change course was their swift denunciation of the latest tone-deaf comments by Mitt Romney, who little more than a week ago they were all trying to help elect president.

In a conference call with campaign donors on Wednesday, Romney blamed his loss in part on “gifts” that a “very generous” President Obama had given to African Americans, Hispanics and young people. It was similar in sentiment to his earlier suggestion — also to a group of wealthy contributors — that 47 percent of the American public consists of government-dependent deadbeats who view themselves as victims.

Asked about Romney’s latest comments, Louisiana Gov. Bobby Jindal bristled and told reporters at a Republican Governors Association meeting here: “I absolutely reject that notion, that description.”

“We need to stop being a dumb party, and that means more than stop making dumb comments,” added Jindal, the RGA’s incoming chairman and a rising star in the party.

The need to reorient and rebuild the party was a major topic of conversation at the governors’ meeting. Among the top concerns was the party’s failure to attract Hispanics, the fact that its voter turnout operation did not live up to expectations, its flatfooted response to Obama’s attacks on Romney and its misplaced optimism that Romney would win.

At one session, former Mississippi Gov. Haley Barbour laid out the need to take an ungentle approach to fixing those problems: “We’ve got to give our political organization a very serious proctology exam. We need to look everywhere.”

Jindal and other governors insisted that putting the party back on track does not mean betraying its traditional principles.

“In the face of the losses, we do have to make changes,” Jindal said. “We need to modernize our party. We don’t need to moderate our party.”

Added Wisconsin’s Scott Walker, who survived a recall effort earlier this year: “It’s not that our beliefs are wrong. We’re not doing an effective enough job articulating those beliefs.”

He also was critical of Romney’s comments. “We’re the party that helps people find a pathway to live the American dream,” Walker said. “They want to have a chance to live the American dream. They want to have a job.”

Just two years ago, fueled by the insurgent forces of the tea party movement, Republicans took back the House in a midterm election that was viewed as a repudiation of Obama. But the president’s relatively easy victory last week suggests that the gains of 2010 masked deeper problems for the GOP.

Still, Republicans see reason for optimism, particularly at the state level. In January, the number of GOP governors will reach 30 — the highest number either party has claimed in a dozen years.

Some of them are considered to be among the Republicans’ brightest prospects for the 2016 presidential election — a topic that was much discussed outside the formal sessions of the meeting, which was held at the luxurious Wynn Encore casino and resort and attended by a large contingent of lobbyists.

Among the attendees was New Jersey Gov. Chris Christie, on his first trip outside his home state since it was ravaged by Hurricane Sandy. As he made his way through the halls, Christie was frequently stopped by well-wishers and congratulated on his performance following the storm.

Back-to-back presidential losses have often forced political parties to look for a new path.

After losing in 1984 and 1988, for instance, the Democrats moved away from their traditional New Deal liberalism and turned to the “third way” centrism advocated and embodied by then-Arkansas Gov. Bill Clinton.

GOP governors are well positioned to lead a similar movement now, said Craig Shirley, a biographer of Ronald Reagan who advises conservative groups. “They’re going to know sooner than the people in Washington what is politically feasible and viable.”

Added Pat McCrory, who last week was elected North Carolina’s first GOP governor in 24 years: “Politically, I think the power and influence of the Republican Party is at the state and local level. Governors, I think, are going to have more influence on national policy than the White House or Congress.”

As recently as the 2000 election, Republican governors united early around then-Gov. George W. Bush of Texas and played an important role in easing his path to the nomination.

“He was one of us, and he was able to get a lot of governors on board early,” recalled Iowa Gov. Terry Branstad.

But Branstad said he was not certain whether Republican governors would form such a phalanx leading up to 2016.

“At this point, people are just trying to analyze what happened” in the most recent election, he said.

While some defeated presidential candidates remain influential figures in their parties, Republicans appear ready to treat Romney as a dinner guest who has stayed too long after coffee.

“There is no Romney wing in the party that he needs to address,” said Ed Rogers, a longtime Republican strategist. “He never developed an emotional foothold within the GOP, so he can exit the stage anytime and no one will mourn.”

Added Branstad, whose state will hold the first presidential nominating contest in 2016: “We’ve got [Florida Sen.] Marco Rubio coming for my birthday on Saturday. We’re going to turn a page.”