CNN

Watch CNN's Wolf Blitzer interview Sen. DeMint Tuesday.

WASHINGTON (CNN) - South Carolina Sen. Jim DeMint reiterated his opposition to the Senate immigration bill Tuesday and told CNN that President Bush might undermine his standing with the American people if he continues to support the measure.

"If he continues to push this bill in spite of the way the American people feel about it, I think that's more likely to undermine him than to say 'we've listened, we're going to take this step by step, and maybe come back next year and look at how to deal with the 12 million who are here after we've proved that we're going to move ahead with real enforcement measures,'" DeMint told CNN's Wolf Blitzer.

DeMint spoke after the president made a rare visit to Capitol Hill in hopes of convincing some GOP senators to support the bill. The senator added that the meeting was "warm," but brushed aside the president's recent prediction that the measure will ultimately pass.

– CNN Ticker Producer Alexander Mooney