Sen. Bernie Sanders, I-Vermont, on Tuesday rejected the idea that he's now the frontrunner in the Democratic race for president despite rising poll numbers.

In an interview with CBS News' Nancy Cordes, Sanders was asked whether he now considers himself the frontrunner after new polls show him ahead of Hillary Clinton in Iowa and New Hampshire.

"No, no," he said. "Look, Hillary Clinton has a very formidable political organization. She has more money than we have. I think both states are going to be very tight," Sanders said in an interview from Capitol Hill.

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Sanders said he's confident he can win the Iowa caucuses on Feb. 1 and the New Hampshire primary on Feb. 9 if people who were not previously active in politics show up to the polls.

"I think if we can create a situation where there's a large voter turnout, I think we're going to do very very well," Sanders said.

The presidential contender said recent attacks on his campaign are "an indication that the Clinton campaign is getting very, very nervous" after months of ignoring Sanders' progress.

"Well, guess what? They're not ignoring us right now. That's for sure," he said.

Sanders responded to comments from Clinton in which she has criticized either part of his platform or voting record. Clinton, for example, said Sanders' Medicare-for-all plan is risky, but Sanders insisted it isn't and would save middle class families "thousands of dollars a year on healthcare."

On Monday, Clinton called on Sanders to commit to repealing a gun manufacturer liability law that he voted for in 2005 that shields them from certain lawsuits. CBS asked Sanders whether gun manufacturers should be held accountable for gun deaths.

"Of course not, that doesn't make any sense," he said, but clarified that if guns fall into the hands of criminals, "of course you hold the gun manufacturers liable."

On Clinton's new proposal for a 4 percent income-tax surcharge for wealthy Americans, Sanders said "it doesn't go anywhere near far enough."

"If we're going to keep up with the rest of the world and have paid family and medical leave," he said, "Her proposal barely covers half of that."

Asked to react to Vice President Joe Biden praising Sanders in a new interview, Sanders said he appreciates his remarks.

"I think he knows that for many, many years, I have been talking about the grotesque level of income and wealth inequality in this country," he said. "He knows how long I've been passionate about that issue."