Democratic presidential nominee Vice President Kamala Harris is embracing some of the Republican Party's old standbys as her own as she gains ground on her GOP rival, former President Donald Trump, pollster Frank Luntz said Tuesday.

Harris "is taking pages out of the Republican playbook," Luntz said on CNBC's "Squawk Box."

The political strategist meant that as a compliment to the vice president.

He credited Harris for what he called her "important use of language," pointing to her focus on small business issues and describing herself as a capitalist. He also noted her party's emphasis on freedom during the Democratic National Convention.

"If you just listen to these general statements, she actually sounds like a moderate Republican," Luntz said.

"So the challenge for the Trump campaign is to just stop with these incessant character attacks, and challenge her on the specific issues and her claims," he said.

"Because, let me tell you something, going to The New York Times' most recent poll, she is beginning to cut through," he added.

That poll from The Times and Siena College, released Tuesday, found Harris leading Trump nationally for the first time since President Joe Biden dropped out of the election as the presumptive Democratic nominee in mid-July.

The poll of 3,385 likely voters surveyed from Sept. 29 to Sunday gave Harris a slight edge of 49% to 46% over Trump, within the poll's margin of error of plus or minus 2.4 percentage points.

The prior Times/Siena poll, in mid-September, found the candidates tied at 47%.

The new poll indicated that Harris is making gains among older voters and Republicans.

And for the first time in the Times/Siena poll, more respondents identified Harris as the candidate representing change instead of Trump.

Luntz said he has seen similar shifts toward Harris in focus groups of undecided voters that he tracks.

He said those voters "moved away from Joe Biden in an extensive way" before the president quit the race.

But, "They move towards Kamala Harris after that first debate," Luntz said, referring to her Sept. 10 showdown with Trump.

"And right now they're asking: Who's telling me the truth?"

Luntz also weighed in on Harris' appearance on CBS News' "60 Minutes," which aired Monday night.

It "took a while," but Harris is now "taking the tough interviews" and "being challenged," he said.

Harris has been criticized for ducking unscripted public appearances and avoiding major media interviews since she took over the Democratic ticket.

But her "60 Minutes" interview flips that script, Luntz argued, because Trump backed out of a scheduled appearance on the same program and has refused to debate Harris again.

"It is Donald Trump who is ducking debates," Luntz said. "She gets credit for that."

Trump's campaign claimed at first that negotiations with "60 Minutes" fell through because the show insisted on doing live fact-checking of Trump's claims.

The campaign later insisted that Trump deserved, but never received, an apology from the show's reporter Lesley Stahl for how she handled his prior appearance in 2020.

Luntz faulted Harris for not offering specifics in her "60 Minutes" interview on her claim that she would partially fund her economic plans by making sure the richest Americans "pay their fair share in taxes."