WASHINGTON >> Turning their attention to the 2012 presidential election, Republican leaders are digging in for a battle over control of the Republican National Committee, judging that its role in fundraising, get-out-the-vote operations and other tasks will be critical to the effort to topple President Barack Obama.

Some senior party officials are maneuvering to put pressure on Michael Steele, the controversial party chairman, not to seek re-election when his term ends in January or, failing that, to encourage a challenger to step forward to take him on.

So far, the effort has been tentative, with Steele’s most ardent opponents working behind the scenes to persuade an alternative to run against him — fearful that any overt moves will create a backlash in Steele’s favor among the party committee members who vote on the chairmanship and tend to view the establishment in Washington with suspicion.

One man leading the effort is a Mississippi Republican Party committeeman, Henry Barbour, who is a nephew of Gov. Haley Barbour of Mississippi — a former chairman of the Republican National Committee himself. Haley Barbour is said by people involved in the discussions to be among those eager to see a change at the top the party and recently criticized party fund-raising under Steele.

Officials close to the presumed new House speaker, Rep. John Boehner, and the Senate minority leader, Sen. Mitch McConnell of Kentucky, said that both men would prefer a new chairman, as well, but that they were also resigned to Steele’s continued leadership should no clear alternative emerge to defeat him.

Several officials — including some directly involved in the discussions — said Henry Barbour and allies of his had approached several potential candidates, including a member of Steele’s “kitchen cabinet” of advisers, Reince Priebus, who is chairman of the Republican Party of Wisconsin and who helped manage Steele’s first election for the chairmanship two years ago.

These officials, all of whom requested anonymity to share details of the talks, said Priebus had recently warned Steele that a run for re-election could prove difficult this time around and advised him to consider leaving the chairmanship at time when he could point to big Republican gains nationwide.

But Priebus has made it clear that he was personally uncomfortable with the idea of challenging Steele directly for the post, given their friendship.

The effort to woo Priebus was first reported Tuesday on the website of The Washington Post.

Steele, who did not respond to interview requests Tuesday, has said he has not decided whether to seek re-election when his two-year term expires in January. The 168 members of the Republican National Committee will convene then in Washington for their winter meeting.

But he is taking steps to build the support he would need in the face of deep opposition by Republican leaders on Capitol Hill and beyond — making personal appearances and granting party money in visits across the country and even in territories like Guam and the Virgin Islands.

“Whether I run or not, I’m going to be judged by what’s happened over the last two years,” Steele told reporters last week, arguing that the party’s early investments in important states and races helped Republicans reach their historic victory of picking up at least 60 House seats. “I think you can safely say the party has re-emerged. It is a very different party. I think it’s a transcendent party.”

Steele conceded his tenure had not been perfect, saying, “Everybody has a learning curve, and clearly I had mine.”

The moves against Steele are a result of a perception that has been developing for months among the party’s seasoned political hands in Washington. His critics say Steele has performed poorly at the helm They argue that his fundraising was lackluster and point to comments he made that at times proved distracting and were at odds with Republican orthodoxy, as when he said the war in Afghanistan was “not something the United States had actively prosecuted or wanted to engage in.”

Steele said last week that he had modeled his effort after the 50-state strategy conceived by Howard Dean when he was chairman of the Democratic National Committee. Dean also infuriated leaders of his party four years ago by directing money and staff members to every state, not simply places where Democrats had a stronger chance to win. He ultimately earned the support of many local Democratic leaders.

Steele dismissed criticism that he has not been a good steward of the party’s finances, saying he had purposefully focused on building grass-roots Republican activism rather than courting high-level donors.

But those working to remove him say a focus on high-level donors is one of the most important roles for a party chairman — if not the most important role — in a presidential election cycle.

Saul Anuzis, who represents Michigan on the Republican committee and ran for party chairman two years ago, said Tuesday that he was considering running again.

“There clearly have been many major donors who have dropped off and have not contributed,” Anuzis said. “That’s a problem.”

Another person approached about the job, the chairman of the Connecticut Republican Party, Christopher Healy, said, “There’s a lot of concern about the state of the RNC and whether current management is up to the challenge.”

But Dick Wadhams, chairman of the Colorado Republican Party, voiced tentative support for Steele. He said that while the national party gave less to states than it had in recent years, the resources it did provide proved critical in his state.

“I don’t have any complaints,” Wadhams said. “I think there’s a chance he could be re-elected.”