At the very least, they say, Mr. McCain remains bruised over Mr. Hagel’s decision not to support Mr. McCain when he became the Republican presidential nominee in 2008, and over a trip Mr. Hagel took with Mr. Obama to Iraq the same year.

“He was very angry about it,” said one of Mr. McCain’s 2008 advisers, who asked not to be identified discussing the complicated dynamics between the two. Mr. McCain “takes policy disputes very, very personally,” the adviser added. He described Mr. McCain’s current view of Mr. Hagel as one of “profound disappointment.”…

Friends say the strains between the two began in 2002, when Mr. Hagel emerged as an early and acerbic Republican skeptic to the Bush administration’s plans for invading Iraq. Mr. Hagel voted for the resolution that authorized the invasion but rapidly became a critic of the Bush administration’s execution of the war. Mr. McCain was equally critical, but he saw the solution in an addition of more than 20,000 American troops, which Mr. Hagel opposed…

Others hold out the possibility of a rapprochement, however remote. “You have two guys who are hurt, and you know how guys are, they don’t make up unless there’s a woman around who forces them,” said one of Mr. Hagel’s former staff members who did not want to be identified discussing the conflict. “They would rather be friends than not, I’m quite certain of that.”