Support for beginning the impeachment process against President Trump has surged over the last week — and yet voters aren’t quite sure the president has done anything illegal.

Even more confounding is that voters still give Mr. Trump among his best job approval ratings since he took office in January 2017.

It’s an odd dichotomy that pollsters labored to explain, and it could go a long way toward determining whether House Democrats ultimately take the next step of drawing up articles of impeachment against the president.

The latest survey from CBS News on Sunday showed 55% of Americans support House Speaker Nancy Pelosi’s move last week to launch an impeachment inquiry, after new allegations that he sought to rope Ukraine’s government into investigating Democratic presidential candidate Joseph R. Biden.

But only 41% say the president has broken the law with what he did.

GOP pollsters say those numbers are likely inflated against Mr. Trump. CBS polled residents rather than likely or even registered voters — the categories that matter when it comes to whether an issue has power at the ballot box.

Still, there’s little doubt the reluctance to contemplate impeachment, which long reigned in polling, has turned into a willingness to consider it.

Morning Consult/Politico’s tracking survey late last week, after the revelations, found 43% say it’s time to begin — up from 36% just a week earlier.

Support among Democrats grew by 13 percentage points, to about four out of five. But so did support among independents, who just a week ago were decidedly opposed, by a 13-point margin. Now they are in favor, 39% to 36%.

Even among Republicans, support for impeachment has grown, though at 10% it’s still not much of a factor.

Matt McDermott, a Democratic pollster, said Mrs. Pelosi’s reluctance to back impeachment had kept a lid on public support. Now her embrace has brought Democrats and some independents along.

“Not to be cliche, but voters are motivated by leaders who lead,” he said. “They want to feel that they’re making the ‘right choices’ politically. These types of cues from political leaders are important, and explain why we’re already seeing perception shifts around impeachment following the support of a formal inquiry by House leaders.”

He said particularly worrying for the GOP was the rise in support among independents. He said Republicans had expected independents to balk at impeachment, but there’s no sign yet of such a backlash.

Republican pollsters questioned the sampling decisions of some of the new polls, but also said they’re not surprised — nor worried — by the new impeachment support.

GOP pollster Michael McKenna said the impeachment question is essentially now the equivalent of asking whether Mr. Trump should be reelected.

He said there are still some voters who view impeachment seriously, but this near to the start of voting in Democratic primaries, for most the impeachment question is about whether to retain Mr. Trump in office rather than a look at whether he’s guilty of high crimes and misdemeanors.

“They may as well, when you ask about impeachment, ask ‘President Trump yes or no,’” Mr. McKenna said. “This is all about the campaign.”

Jim McLaughlin, a Trump campaign pollster, points to a survey he did earlier in September, before the latest impeachment developments, that asked whether voters wanted to see congressional Democrats focus on impeachment or legislating. Two-thirds said legislating.

He said that’s a good place for Mr. Trump to be — and it’s showing up in those job approval numbers, which are higher than they’ve been since the immediate days after he took office in 2017. In national polling, Mr. Trump is at about 45%, maintaining the level he’s been at since a bump after his State of the Union speech.

Mr. McLaughlin says his surveys are even better: “We’ve consistently had him about 47, 48% job approval.”

He said that rising approval spurred Democrats to go for impeachment. He also said there could be an element of preemption in that Democrats are worried about what the Justice Department’s investigation into the Obama administration’s handling of the 2016 election will show, and impeachment is a preemptive strike.

“Once they saw the president’s numbers going up, they said ‘We gotta do something,’ so they made this thing up,” he said.

He predicted the impeachment push will serve to unify Republicans heading into 2020, consolidating Mr. Trump’s backers.

The millions of dollars the Trump campaign raised in the hours after Democrats announced their impeachment inquiry was the first tangible sign of that.

In CBS’s poll, 59% of Republicans said Mrs. Pelosi’s move makes them want to defend Mr. Trump. But a surprisingly strong 34% said they’re looking to see what develops. Just 7% suspect Mr. Trump of wrongdoing.

⦁ Alex Swoyer contributed to this report.

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