The extraordinary pressure campaign, taking place in public and private, is forcing the party’s senators to choose between their loyalty to the Intelligence Committee and to the president’s family as it attempts to quash any remaining investigations of the president after the completion of the investigation by the special counsel, Robert S. Mueller III.

It also put Mr. Burr and the Intelligence Committee on their heels. After two years of conducting the only bipartisan congressional investigation into Russia’s election interference campaign, the committee is in the final stages of its work and had hoped to avoid partisan fireworks that would distract from the substance of its final warnings about the Russian threat.

Even as the chairman privately defended his approach, the attack appeared to be paying at least some dividends for the president’s eldest son, who could theoretically face calls to be held in contempt of Congress if he does not comply with the subpoena. (Several people close to the younger Mr. Trump said he was unlikely to appear in person and said one option was to invoke his Fifth Amendment rights in a written response.)

For Mr. Tillis, who is not on the Intelligence Committee, the squeeze might feel familiar. Earlier this year, he publicly came out against Mr. Trump’s declaration of a national emergency to secure funding for a border wall against Congress’s wishes. But in March, facing criticism from home-state conservatives, he flipped and voted against a resolution of disapproval for the emergency declaration.