Sid Rosenberg, an outspoken radio host in New York City and an ardent supporter of President-elect Donald J. Trump, was at a loss.

Surely, he told Mr. Trump’s daughter-in-law, Lara Trump, the president-elect understands that some of New York City’s problems can be tied to Mayor Eric Adams.

“He can’t be happy with what’s going on in New York City: the crime, the filth, the dirt, the illegals,” Mr. Rosenberg said earlier this month while interviewing Ms. Trump on his morning talk show. “Yet almost every day in the news, we hear in New York about this blossoming relationship between your father-in-law and the mayor.”

“And Trump supporters in New York don’t understand,” Mr. Rosenberg said.

The spectacle of the Democratic mayor of New York City and one of the most divisive Republican presidents in modern American history making overtures to each other has been the talk of New York City’s political class. But it’s not just Democrats who are upset.

Amid the Republican Party’s unification around Mr. Trump, an early crack has emerged in his hometown. Some New York Republicans who disdain Mr. Adams’s governance of New York City are now watching with concern as the leader of their party makes nice with a Democratic leader who once proudly called himself the “Biden of Brooklyn.”

The latest example came Monday, when Mr. Trump said he would consider pardoning Mr. Adams, who was indicted on federal corruption charges in September, saying that the mayor, like himself, was the victim of prosecutorial overreach. “He was treated pretty unfairly,” Mr. Trump said.