Senate Republicans unhappy with the leadership of the Republican National Committee are quietly pushing for change in the wake of Chairman Michael Steele opting to run for a second term.

Congressional Republicans have little influence over the RNC leadership elections. That factor, and a desire to avoid inserting the heavy hand of Washington into a contest mostly decided by 168 RNC committee members in the states, has led many Senate Republicans to keep out of the debate over Steele.

But two key Senate Republicans on Tuesday expressed what many in their Conference have been feeling throughout the chairman’s controversial two-year tenure.

“I’m looking for what the alternatives are,” said Sen. Jim DeMint (S.C.), who raised millions of dollars for conservative Senate candidates last cycle and is flirting with running for president in 2012. “I appreciate his service, but 2012 is real important.”

Senate Republican Policy Committee Chairman John Thune, who is the fourth-ranking Senate Republican and is actively considering a presidential bid, also suggested during a brief interview that he would like to see a new RNC chairman assume office next year.

The South Dakota lawmaker did not specifically call for Steele’s defeat in the Jan. 14 election, but he said his party’s national committee needs more effective fundraising and strategic management going forward. Steele has endured withering criticism on both fronts from the outset of his taking the job, and even that didn’t come easy since he only won the race after six rounds of balloting eliminated the competition.

“Those of us who are not members of the committee don’t have a vote in this situation. But I think that we obviously have a stake in the outcome,” Thune said. “I think we want to make sure that whoever leads the RNC in the next couple of years is capable of raising the resources that will be necessary to make sure we can run winning campaigns  — and I think that’s the issue, probably, that [Steele] will have to answer about.”

A similar Steele critique has been widespread among Republicans at both ends of Capitol Hill, particularly within GOP leadership.

Many Republican political operatives have complained that the GOP’s victories Nov. 2 could have been even greater had Steele done a better job of fundraising and overseeing the RNC’s crucial ground-game activities, rather than parading around the country in a large red bus with the words “Fire Pelosi” emblazoned on the sides.

One Republican involved in a statewide 2010 campaign called Steele an “absolute disaster” who cost the GOP victory in multiple statewide races. “The turnout operation in my state was so amateurish and so underfunded, we never had a shot,” this Republican said.

But Steele, the former Maryland lieutenant governor and first African-American to lead the committee, is not without his fans. His tenure has featured historic Republican victories at all levels.

On Nov. 2, the GOP picked up a net gain of 63 House seats — the largest Republican gain in more than half a century — six Senate seats and a nearly 700 state legislative seats. Earlier in 2010, the GOP won an additional Senate seat in a special election in deep-blue Massachusetts, and in 2009, the Republicans won Democratic-held governorships in New Jersey and Virginia, stats that Steele made sure to mention when making his surprise announcement Monday night.

Sen. Orrin Hatch, who is up for re-election in 2012 and has been among the most avid fundraisers on behalf of the National Republican Senatorial Committee since 2007, said he has been pleased with Steele’s leadership of the RNC. In praising Steele, the Utah Republican alluded to criticism that Steele has received for his flamboyant, media-friendly approach versus the typical behind-the-scenes managerial role assumed by most chairmen.

“I like Michael Steele a lot, and I think he tried his dead-level best,” Hatch said, describing Steele as “out there, out front, leading the charge.”

Unlike most incumbents, Steele starts this race as the prohibitive underdog.

Among the contenders he will be challenging in next month’s election are former allies, and some of the RNC members he counted as supporters have already endorsed his rivals in the contest. At least one of his rivals from his first bid, Saul Anuzis of Michigan, is running again.

News outlet estimates have placed Steele’s support in the 2011 vote at about 40 to 50 members, about half of what he would need to win. His entry means the other candidates will divide up the anyone-but-Steele vote.

The top three favorites, according to a knowledgeable Republican source, are former RNC official Maria Cino, Wisconsin Republican Party Chairman Reince Priebus and former Ambassador Ann Wagner of Missouri. Cino has deep support inside the Beltway, which is why some Republicans give the nod to Priebus and Wagner.

Priebus managed Steele’s race for chairman in 2009 and served as general counsel for the RNC until earlier this month. Norm Semanko, chairman of the Idaho Republican Party and a Steele ally, took his place.

Another former Steele ally, Gentry Collins, created a splash in November when he resigned as political director of the RNC. He wrote a letter detailing problems at the RNC during Steele’s tenure, and on Monday he launched his own bid for chairman. He cited as assets his experience as a political operative in Iowa and at the RNC. He announced that Iowa GOP Chairman Matt Strawn, who announced last week that he will run for a second term, and Connecticut GOP Chairman Chris Healy, who recently abandoned the idea that he would run for chairman himself, are supporting him.

Former Sen. Norm Coleman flirted with running, but a spokesman for the Minnesotan confirmed Tuesday that Coleman would stand by his commitment not to seek the RNC chairmanship if Steele ran for re-election.

The RNC declined to comment, and an e-mail requesting comment sent to Steele’s re-election team elicited no response, but Steele made his case Monday night in a letter to RNC members.

“Our work is not done; and my commitment has not ended,” he wrote. “I believe the worst thing we can do now is to look backwards. Who you elect as our next Chairman will speak volumes about our willingness to truly be the party of Lincoln. What we do as a party must reflect the realities of the political marketplace and the voters. Growth for our movement requires new allies.”

Tricia Miller contributed to this report.