Correction: An earlier version of this story incorrectly referred to Rep. Ron Paul as the only military veteran in the GOP presidential field. Texas Gov. Rick Perry also is a veteran.

Ron Paul is the Republican whom establishment Republicans want to ignore. But right now they can’t seem to take their eyes off him.

Many in the GOP consider him a fringe presidential candidate, a politician whose views on foreign policy and the legalization of drugs put him far outside the party’s mainstream. They would like to dismiss him, but here in Iowa it’s impossible to do so.

As he has for weeks now, Paul drew strong crowds on Thursday at a trio of town hall meetings in the western part of the state.

“We are feeling pretty good about the way things are going,” he said at a midday rally in the little town of Perry, where he delivered his small-government message to a crowd of about 100. “The momentum seems to be building and we are very pleased.”

Among the Republican candidates, only Mitt Romney seems comfortable with the strength Paul is showing. That’s largely because the former Massachusetts governor could rebound from a loss in Iowa with a victory in New Hampshire.

The others cannot count on that. Until they show that they can win more votes than Paul, they will have to contend with him as well as Romney. In that sense, Paul is an obstacle to those who hope to enter into a one-on-one post-Iowa matchup against the former Massachusetts governor. Also, if Paul wins or finishes second to Romney, whoever came next would have to compete with him for much-needed attention.

So the others are going after him. Former senator Rick Santorum (Pa.), rising in polls in Iowa at just the right moment, said Paul, a congressman from Texas, has no track record of accomplishment on Capitol Hill. He said that Paul may talk big but that he can’t deliver on anything.

Rep. Michele Bachmann (Minn.), embarrassed by the defection of her state chairman to Paul’s campaign Wednesday, has been hammering the congressman for weeks. She told reporters Thursday that he is dangerous on Iran, weak on same-sex marriage and plain wrong on drugs.

Romney has generally refrained from criticizing Paul, but even he said this week that his rival is incorrect on Iran. But he probably won’t go much further in trying to knock down the congressman. He’d like nothing better than a final contest between him and Paul, rather than, say, former House speaker Newt Gingrich (Ga.) or one of the others.

That’s why the “super PAC” supporting Romney is focused here on Gingrich, with a new ad out Thursday. This strategy is reminiscent of what happened in 1996, when conservative commentator Pat Buchanan and his “peasants with pitchforks” were charging toward victory in the New Hampshire primary over front-runner Bob Dole.

Dole’s team ignored Buchanan, knowing he was not a long-term threat, and used its TV ads in the final days before voting to attack a rising Lamar Alexander (then a former governor and now a senator from Tennessee). Buchanan won and went nowhere. Alexander finished third and was no longer a concern.

Romney and his allies are playing the same game now. The others can’t afford to. Paul is someone they have to deal with. But they are confronting a rival who is nothing if not unusual and whose supporters represent a coalition unlike anything anyone else can attract.

Paul is the candidate who rails against military intervention abroad but is one of only two veterans in the field and the only candidate in these last days before the Iowa caucuses to hold a salute-to-veterans rally, which drew a big and boisterous crowd on Wednesday night.

He denounces Wall Street bailouts and drew a handful of protesters from the Occupy movement at Wednesday’s rally.

His campaign speeches are more tutorials than political oratory. He talks about obscure economic theories. He casually tossed into a speech Wednesday the fact that there are 371 grains of silver in a silver dollar. Which other Republican even knew that?

Paul was tea party before “tea party” was in the modern political lexicon — tea party when tea party wasn’t cool. He sees his rivals as cut from the same establishment cloth. “If you pick another status-quo candidate, nothing is going to change,” he says.

He wants to shrink government greatly and said Wednesday that about 80 percent of what the government does is “technically unconstitutional.”

Some supporters are drawn by the quirky positions he holds, the social libertarianism he espouses. What brings many others to his candidacy is that he presents the purest version of constitutionality. The appeal is straightforward: “Why don’t we just follow the Constitution?”

Paul’s support may have a low ceiling, given the nature of his views. That has been the theory that his rivals have long assumed. He received just 10 percent of the vote in Iowa four years ago but is polling at about double that number now. He may not be able to get more than 25 percent. But with the field closely bunched, even a candidate with a ceiling can have a significant impact.

Paul could prove to be nothing more than a passing storm for the Republican Party. If Romney were to win here Tuesday and then in New Hampshire, he would be in a strong position to capture the nomination. In that case, attention could shift quickly from Paul’s candidacy. But his coalition includes many conservatives whom the eventual nominee will need in the general election.

No one would have predicted a year ago that the Texas congressman would receive the kind of attention he is garnering now. What the party does about him will be a challenge ahead.

Staff writers Amy Gardner and Nia-Malika Henderson contributed to this report.