“Some of this is a feeling that it’s time for new blood,” said Brendan Steinhauser, an organizer with FreedomWorks, a national group that has worked with Tea Party groups on several primary challenges.

Mr. Lugar said at a breakfast with reporters this month that he believed that many Tea Party supporters were motivated by anger “about how things have turned out for them.” They want to express themselves, but their complaints often boil down to nothing more specific, he said, than “we want this or that stopped, or there is spending, big government.”

“These are all, we would say, sort of large cliché titles,” he said, “but they are not able to articulate all the specifics.”

The advocates in Indiana, which national Tea Party groups say has the most organized of the primary efforts, point to Mr. Lugar’s push for the New Start nuclear treaty, which the Senate approved in December; his sponsorship of the Dream Act, which would grant a path to citizenship for limited groups of illegal immigrants; and his votes for President Obama’s picks for the Supreme Court, Justices Sonia Sotomayor and Elena Kagan.

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“The senator would call it bipartisanship, but we think you’re siding with the other side,” said Greg Fettig, a Tea Party supporter in Indiana.

Another, Mark Holwager, said, “He may have been a conservative at one time, but he definitely leans to the left now.”

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The coalition of Tea Party groups, calling itself Hoosiers for a Conservative Senate, plans to hold a caucus in June where the 70-odd groups involved will choose a candidate to run against Mr. Lugar in the primary next May. In the meantime, the group has designated a coordinator for each of the state’s Congressional districts to begin a campaign to educate voters about what Tea Party supporters call Mr. Lugar’s liberal record.

The group has also had discussions with several national groups that played a role in primaries last year where establishment candidates or Republican incumbents lost to Tea Party challenges, including FreedomWorks, the Tea Party Express and the Club for Growth.

Those behind Tea Party challenges say they learned their lesson about splitting the vote from several primary contests last year, including the Senate races in Illinois and Indiana and Congressional races in Virginia, where a flood of Tea Party candidates resulted in a moderate or establishment candidate winning.

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But so far there are few declared candidates. In Utah, some Tea Party supporters say they would back a run by Representative Jason Chaffetz, who won his Congressional seat in 2008 after challenging an incumbent Republican, Chris Cannon, in the primary. That race became a kind of trial run for Tea Party primaries, with many of the same people who worked on Mr. Chaffetz’s campaign working to unseat Mr. Bennett last year.

Still, there is some division. The Tea Party Express, a national group started by longtime Republican consultants, recently announced that it would not back a challenge to Mr. Hatch, calling him “as good as it gets” for Republicans. The Club for Growth, which has poured money into other Republican primary challenges, issued a statement disagreeing.

Utah is a reliably Republican state, so whoever wins the primary is almost assured to win the general election.

In Maine, even some Tea Party supporters say the challenge is trickier: voters have long preferred moderates and independents. Still, some note the success of the state’s new Republican governor, Paul LePage, who was elected with Tea Party support and is winning applause for pushing to cut the budget.

“The way the state’s been turning, I think if we got a decent conservative in there, it wouldn’t be that much of a problem to win,” said Pete Harring, a Tea Party supporter in Maine. “People are starting to realize we’re just too deep in a hole; we have to do something.”

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Mr. LePage narrowly beat an independent candidate in a five-way general election, winning less than 40 percent of the vote. With that in mind, Tea Party supporters have also discussed running a third-party candidate.

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In Indiana, several Tea Party supporters met with Mr. Lugar last month, and he argued his conservative credentials. Unconvinced, they announced that they would pursue a primary challenge, and that the first step would be to unify behind one Republican. Potential candidates include a state senator, Mike Delph, and the state treasurer, Richard E. Mourdock.

At the meeting this month, the Tea Party organizers signed a letter that “with deep gratitude and respect” asked Mr. Lugar to resign. With the rise of conservative awareness in America, “the emergence of the modern day Tea Party, and your own more social-liberal perception on issues, we find ourselves at odds,” they wrote.

Mr. Lugar won his last term with 87 percent of the vote after Democrats declined to challenge him. He says he intends to run aggressively, and not change his positions.

“A lot of conservatives believe you have to kowtow to the Tea Party,” said his spokesman, Mark Helmke. “We reject that premise.”

Mr. Holwager argued that there is a disconnect between Tea Party supporters and many of their representatives in Washington.

“Heartland America doesn’t feel the same way as people in the cities,” he said. “We do believe in religion, we go to church all the time, we shoot and fish, and love our families. Some of the time you wish folks in the cities would come live with us and see how we live.”