The change in Mr. Bush’s tone is a calculated strategy, interviews show, with two different but crucial aims now that Mr. Trump is proving to be a long-term obstacle, not a passing nuisance: to dilute Mr. Trump’s right-wing support by proving that he is not a genuine conservative and to show a wary Republican Party that Mr. Bush is enough of a street fighter to survive a nasty nomination contest.

In a frank if grudging acknowledgment of the altered political landscape, Mr. Bush on Monday proclaimed Mr. Trump “a serious candidate.”

“He needs to be held to account for his views,” he told reporters at a restaurant near the border.

The new approach, which began with Mr. Bush’s remarks last week in New Hampshire and then on Monday, may soon expand, drawing in surrogates who can broaden the critique beyond Mr. Trump’s policies — much the way that Mitt Romney sought to undermine rivals like Newt Gingrich in 2012, said people familiar with the discussions, who insisted on anonymity to discuss plans that were not set in stone.