NASHUA, N.H. — The big question facing Mayor Giuliani heading into tomorrow’s primary here is not so much whether he will win, but whether he can finish ahead of Rep. Ron Paul.

Mr. Paul, the stridently anti-war Texas congressman, nearly tripled Mr. Giuliani’s vote total in Iowa and is running neck-and-neck with him in New Hampshire, with each hovering around the 10% mark in the polls.

For Mr. Giuliani, the undercard bout with Mr. Paul is the starkest sign yet of how far his star has fallen in recent months. In a debate last May, Mr. Paul served as the foil for what may have been the highlight of Mr. Giuliani’s campaign, when the former mayor demanded that he apologize for suggesting that America’s presence in the Middle East had caused the terrorist attacks of September 11, 2001.

While Mr. Giuliani was then the undisputed Republican frontrunner, his campaign is now treading water, looking for solid but unspectacular performances in the next three primaries and to stay afloat until the vote in Florida in three weeks.

Though analysts and even some Giuliani supporters are skeptical of that strategy, nearly all agree it is the only option he has left. But what is largely left unsaid is that finishing behind Mitt Romney and Senator McCain is one thing, while losing twice in a row to a candidate once considered the GOP fringe’s equivalent of Rep. Dennis Kucinich is quite another. “That would be embarrassing,” an associate professor of political science at the University of New Hampshire, Dante Scala, said.

In interviews after Saturday night’s debate in Manchester, advisers to Mr. Giuliani said they placed no particular importance in beating Mr. Paul here. But the campaign manager, Michael DuHaime, did add Mr. Paul to the handful of GOP hopefuls considered “serious candidates.”

“He gets a chunk of votes,” Mr. DuHaime said, “and anyone who gets a chunk of votes deserves to be taken seriously in terms of how they weigh in the race.”

The advisers stuck to the campaign’s message of saying that Mr. Giuliani’s goal for New Hampshire was a “strong finish,” without defining what that entails. One top supporter, however, did offer up a number. “The top four,” said Rep. Peter King of Long Island, a surrogate traveling with Mr. Giuliani in New Hampshire. “It keeps him competitive.”

Based on the current line-up of Republicans in the state polls, it also keeps him above Mr. Paul, with Mr. McCain, Mr. Romney, and the Iowa winner, Michael Huckabee, ahead of him.

In a CNN/WMUR poll released yesterday, Mr. McCain was leading with 32%, followed by Mr. Romney with 26% and Mr. Huckabee with 14%. Mr. Giuliani edged Mr. Paul, 11% to 10%. A USA Today/Gallop poll, also released yesterday, showed Mr. McCain leading with 34%, while Messrs. Giuliani and Paul were tied at 8%.

A weak finish in New Hampshire could be more damaging to Mr. Giuliani than it was in Iowa because he has made a much more aggressive effort here. He is running television ads in the state, which he did not do in Iowa, and he acknowledged in a news conference yesterday that he has spent as much time in New Hampshire as he has in any other state.

The electorate also seems to set up well for him in New Hampshire, with its prevalence of independent and fiscally conservative voters over the social conservatives who dominate in Iowa. “This is a place where he could have done well, and arguably, should have done well,” Mr. Scala said.

Still, Mr. Giuliani’s campaign is modest here. More than 200 people packed a house party for him yesterday in Hollis, N.H., but it was his only event with voters of the day. He later held a press conference to announce the endorsement of the New England Police Benevolent Association before attending a candidate forum last night.

He refused to predict an outcome, saying he hoped only to do “the very best I can do.”

“My candidacy is an unconventional candidacy,” he said, adding that he had exceeded expectations simply by being in contention at this point in the race — a curious statement from a man considered the party’s front-runner for long stretches of last year.

Speaking at the house party, he kept a positive message, although he did manage to get a dig in at Mr. Paul. Asked about his possible Cabinet, Mr. Giuliani cited President Lincoln, who named party rivals as his top advisers. He then joked that his Cabinet could comprise all of the Republicans on the debate stage Saturday night, “maybe with one exception.” The crowd laughed, though it was initially unclear whether he was referring to Mr. Paul or Mr. Romney, who was roundly criticized at the debate. He later acknowledged it was Mr. Paul, citing his “isolationist” foreign policy views.

Advocating dramatic cuts in the size of government, a strong American dollar, and a foreign policy of nonintervention, Mr. Paul has generated a significant grassroots following, both on the Internet and on the ground in New Hampshire. His fund-raising strength also has astonished the political establishment. He reported raising $19 million in the fourth quarter alone, outpacing all other Republicans.

In the Iowa caucuses, Mr. Paul finished with 10% of the vote, while Mr. Giuliani had 3.5%. It was a result the Paul campaign relished, particularly considering their repeated fights over foreign policy during the GOP debates. “We loved that,” Mr. Paul’s son, Rand Paul, told The New York Sun. He said the campaign “really hoped to beat up on the mayor in New Hampshire, too.”

The Granite State could be even more fertile ground than Iowa for Mr. Paul, who ran for president as a Libertarian in 1988. As his supporters often mention, the state motto is “Live Free or Die.” “They’re more libertarian in that sense. Iowa’s just not the same way,” Rand Paul said.