Privately, some of Mr. Trump’s most ardent defenders worried that Mr. Graham would become yet another Republican committee chairman in the Senate who was failing the president. That is how they regarded Senator Richard M. Burr of North Carolina, the chairman of the Intelligence Committee, after he issued a subpoena in May for the younger Mr. Trump during his panel’s investigation into Russian election interference.

In that case, Republican allies of the president declared war on the Intelligence Committee and turned several Republican senators against Mr. Burr. Some party leaders say that Mr. Graham has reason to worry about the same fate — and that voters might follow suit.

“Republican voters expect elected officials to show their full support of the president,” said Matt Moore, the former chairman of the South Carolina Republican Party. “Thus far, Senator Graham has navigated that dynamic really well. But he’s only one tweet away from being on the president’s naughty list.”