Given how intense the opposition to Mr. Bolton is, nominating him would further complicate what is already expected to be a bumpy confirmation process for Mr. Trump’s State Department team. Mr. Tillerson has no shortage of skeptics in the Senate, many of them Republicans who are preparing to question him exhaustively on his relationship with Vladimir V. Putin, the Russian president.

Still, picking as a No. 2 someone like Mr. Bolton, whose hard-line positions might help neutralize reservations over Mr. Tillerson, is not necessarily enough incentive for Mr. Tillerson’s Republican critics.

“I like John Bolton and hope he gets a senior position,” said William Kristol, the editor of the conservative Weekly Standard and a Trump critic. “But the Trump people shouldn’t kid themselves that any selection as deputy would erase the deep concerns about Tillerson of those of us who believe we can’t afford to continue Obama’s policy of supineness to Putin.”

Though Mr. Bolton, 68, is admired by conservatives like Mr. Kristol who agreed with the Bush administration that American military intervention was a necessary force for promoting stability throughout the world, there are also many Republicans who want to leave the Bush years in the past.

During the campaign, Mr. Trump professed to be one of them. He called the war in Iraq “a big fat mistake” and accused the Bush administration of lying to the American people about Saddam Hussein’s possession of weapons of mass destruction.

Mr. Trump brought up those faulty intelligence assessments again last week when he said “the same people” in the American intelligence agencies who were wrong about the war in Iraq believed that Russia had intervened to tip the presidential election to him.