Though he trails Rudy Giuliani in national polls, Mitt Romney is going on offense in Iowa and New Hampshire, where he is holding the lead in Republican voter preference polls.

The former Massachusetts governor is holding 29 percent of the vote in a Des Moines Register poll of 405 Republican voters, roughly the same number he earned in May. The status of the candidates beneath him appears to be in flux, however. Fred Thompson, who joined the race last month, is in second place with 18 percent. Former Arkansas Gov. Mike Huckabee has also gained ground, with 12 percent, while the former New York mayor has slipped from 17 to 11 percent. Arizona Sen. John McCain has fallen to just 7 percent from 18 percent in May.

Click here to read Des Moines Register polling information.

On the Democratic side, New York Sen. Hillary Rodham Clinton is at the top of the heap, an encouraging sign of progress toward overcoming a big hurdle in the race.

Although the New York senator is the clear front-runner in national surveys, Iowa has remained an elusive prize. She has been in a tight race with John Edwards and Barack Obama in the state that begins the primary campaign voting in three months. She has 29 percent of voters' support in the latest poll of respondents — up from third place in May — while Edwards and Obama are in a statistical tie 23 percent to 22 percent respectively. The Des Moines Register poll of 399 respondents has a margin of error of 4.9 percentage points.

Stumping in Iowa over the weekend, Edwards suggested that he would be better than Clinton at winning over red state voters.

"Democratic voters have a very clear choice between Senator Clinton with both all of the good and bad that comes with her and John Edwards who has actually won in a red state, and who can compete in every single place in America," said the former North Carolina senator.

Clinton, however, is getting big support from the Democrats' most popular leader, her husband, former President Bill Clinton. The two are campaigning together, and Iowa caucus-goers apparently appreciate the company, according to the Des Moines Register poll.

"I'm doing everything I can to earn the support of Iowans," Clinton said during a stop in New Hampton. A standing room only crowd at a community center was warmed up by listening to disco hit "Ain't No Stoppin' Us Now."

"I pay absolutely no attention to what any poll says or what any pundit on TV says," the former first lady said. "I have absolutely no interest in that. Nobody has come to a caucus yet. Nobody has cast a vote yet."

But Clinton hasn't won over everyone in New Hampshire. Randall Rolph of Nashua challenged her for voting last month to designate the Iranian Revolutionary Guard Corps a terrorist organization. Some Democrats said they feared that such a designation could be interpreted as a congressional authorization of military force in Iran.

Rolph compared Clinton's vote on the Iran measure with her vote to authorize war in Iraq. "It appears you haven't learned from your past mistakes," he said.

Clinton responded that his interpretation was wrong and suggested that someone put him up to asking the question. The man said he did his own research and was offended that she would accuse him of getting it elsewhere. She apologized but insisted he must be looking at the wrong version of the bill.

Their exchanged grew heated as he insisted the bill would authorize combat. Clinton snapped back, her voice rising, "I'm sorry, sir, it does not."

"I know what we voted for, and I know what we intended to do with it," she said. She said it gives the authority to impose penalties.

Many in the crowd applauded her in an effort to cut off the exchange, although afterward at least a couple others in the room came up to thank Rolph. He said he is still undecided about which Democrat he will support, but it will not be Clinton.

As for Republicans in New Hampshire, Romney's team is concerned about his numbers. He is now running even with Giuliani in a state he used to lead by double-digits.

The close contest may have led to Romney's attack on Giuliani this week in which he criticized the former mayor's tax and spending record. That led to an angry barrage of e-mails between the two campaigns.

For now, however, Giuliani is hanging onto his lead nationally. The latest AP-Ipsos poll has him at 27 percent, ahead of Thompson with 23 percent. McCain is at 13 percent while Romney takes just 11 percent of the vote.

FOX News' Malini Bawa and The Associated Press contributed to this report.