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Sen. Marco Rubio is losing support among Republicans after the latest debate and heading into Tuesday's first-in-the-nation primary in New Hampshire, a long-time GOP strategist said Monday.

Following Rubio's surprisingly good performance in the Iowa caucuses, the Republican establishment appeared last week to be coalescing around him as the person to challenge billionaire Donald Trump and Sen. Ted Cruz for the GOP presidential nomination, according to former Minnesota Rep. Vin Weber, who supports Jeb Bush.

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But Rubio did not handle the pressure well during the latest debate, said Weber, who has worked on campaigns for such Republicans as Mitt Romney and George W. Bush.

"I think the whole attitude of the Republicans toward the race changed a lot this weekend," he told CNBC's "Squawk Box."

Going into the weekend, most Republicans wanted to narrow the field, establish a clear challenger to Cruz and Trump, and wrap up the contest quickly, he said.

"People are hitting the brakes, saying let's not rush to judgment. Let's get this right. I think there's a greater emphasis on experience," he said.

Republicans now want to see how more experienced candidates fare in New Hampshire and South Carolina, Weber said. That means the Republican battle could last longer than thought last week, he added.

New Jersey Gov. Chris Christie launched a debate night attack challenging Rubio's record that news media focused on throughout the weekend. But the change in sentiment is most likely to benefit Ohio Gov. John Kasich and former Florida Gov. Bush, said Weber.

Rubio has the support of 17 percent of likely Republican voters in New Hampshire, putting him behind frontrunner Trump, who commands 30 percent of that group's support, according to a NBC News/Wall Street Journal poll conducted after the Iowa caucuses.

Cruz trails just behind Rubio with 15 percent, followed by Kasich with 10 percent and Bush with 9 percent.

The doubts raised by the Republican debate might also shake up the Democratic primary, former Michigan Gov. and Hillary Clinton supporter Jennifer Granholm said Monday.

There are many independents in New Hampshire, where voters can decide to cast ballots in either party's primary contest. After this weekend, they may opt to weigh in on the Republican ballot rather than vote in the Democratic primary, she said.

That should benefit Clinton, she added, though she acknowledged Vermont Sen. Bernie Sanders will likely take New Hampshire.

Sanders leads Clinton by 13 percentage points in an average of polls among the state's likely Democratic voters in his neighboring state.