With little hope of catching Donald Trump, Ohio Gov, John Kasich is increasingly taking aim at Ted Cruz, hoping to unseat the Texas senator as establishment Republicans’ go-to alternative in the GOP presidential race.

Mr. Kasich recently labeled Mr. Cruz a “smear artist” in the wake of an ad attacking the Ohio governor in Wisconsin. Ahead of the April 19 New York primary, Mr. Kasich’s campaign also released several ads that specifically targeted Mr. Cruz, hitting the Texas Republican for his line in the run-up to the Iowa caucuses deriding Mr. Trump for embodying “New York values.”

“He’s trying to break up the Cruz narrative that the party is consolidating behind Cruz to defeat Trump,” said GOP strategist Ford O’Connell. “He wants to find a way to remain relevant, and when it comes convention time he can say ‘wait a minute — I’m as viable an option as anyone else.’”

The 95 Republican delegates in play in the New York primary represent the fourth-highest single-state total in the 2016 GOP race, and Mr. Trump has been leading by some 30 points in recent polls — and polling above 50 percent. A Fox News poll released Sunday had Mr. Trump at 54 percent, Mr. Kasich at 22 percent, and Mr. Cruz at 15 percent in the state.

The 50 percent mark is significant, because candidates win all three delegates up for grabs in each of the state’s 27 congressional districts if they win a majority of the vote in an individual district, and the remaining statewide delegates also become winner-take-all if a candidate wins an overall majority statewide.

“If there’s a sense that they’re looking at surveys and Trump, from Kasich’s point of view, is comfortably under 50 percent, then trying to make sure that you’re moving votes away from Cruz so you can move up [would] clearly be the right strategy,” said Republican strategist David Winston. “The threat here is if they’re just swapping votes but Trump’s at 52 percent it is rearranging the deck chairs on the Titanic.”

The primary objective of many Republicans determined to stop Mr. Trump at this point is to find any way to prevent him from reaching the 1,237 delegates he needs to clinch the nomination before the convention. Mr. Kasich cannot get to the 1,237 number before Cleveland, and Mr. Cruz, though he picked up additional delegates in Colorado over the weekend, has a substantially uphill climb.

The latest tally, according to the Associated Press, has Mr. Trump with 743 delegates, Mr. Cruz with 545, and Mr. Kasich with 143.

Some bent on stopping Mr. Trump have argued that energy spent attacking candidates other than the front-runner ultimately serves to boost the billionaire businessman by preventing consolidation behind a single anti-Trump alternative.

But Gerald Benjamin, a political scientist at the State University of New York at New Paltz, said Mr. Kasich’s decision to target Mr. Cruz actually does make sense, and that he didn’t foresee it playing a big role in boosting Mr. Trump’s prospects.

“I think the anti-Trump constituency in the Republican Party is not a pro-Cruz constituency, either,” Mr. Benjamin said. “He’s not going to drive voters from Cruz to Trump.”

Mr. Benjamin said that depending on where Mr. Kasich picks his spots, he could parlay a relatively strong performance in certain pockets of New York into a broader narrative going forward.

“If he does well in the suburbs in Westchester, Nassau County and Suffolk, if he does well in the suburbs of Rochester and Buffalo, he’s essentially creating the argument that he can appeal to the relatively right-center constituency that is the constituency in New York and other highly urban states with major suburban areas,” he said.

That could apply to some of the northeastern states that vote on April 26 like Pennsylvania and Maryland, where Mr. Kasich has also been jockeying with Mr. Cruz for second place.

“We’re going to have to have some momentum and accumulate more delegates, and we will,” Mr. Kasich said Sunday on CBS’s “Face the Nation.” “Our strategy is to continue to move forward to accumulate delegates. And so we’ll see what happens out of New York, Pennsylvania, Maryland, Connecticut — we’ll see what happens there.”

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