Mr. Cruz has begun airing ads in Wisconsin accusing Mr. Kasich of cronyism as Ohio’s governor. He has sent mailers attacking Mr. Kasich’s record on spending and his views on the Second Amendment. And his team has accused Mr. Kasich of auditioning to be Mr. Trump’s vice president, with some joking that he must be receiving some sort of payment.

The exasperation with Mr. Kasich has been long in the making, but as Mr. Trump stumbles and his grasp on the nomination is loosened, it is reaching a boiling point…

“I’m not getting out,” Mr. Kasich said. “Why would I get out, particularly when Trump’s worried I’m going to get his votes?”

Mr. Kasich’s team sees opportunities for delegates in states like New York, Maryland and Delaware, often in suburban areas. Mr. Cruz’s team sees any such gains as assisting Mr. Trump, particularly in New York, where Mr. Kasich’s presence could ensure that Mr. Trump receives a delegate even in congressional districts he loses to Mr. Cruz, given the state’s rules.