About three hours after former President Donald J. Trump was nearly assassinated, on a Saturday evening in mid-July, Robert F. Kennedy Jr. got a phone call: Would he consider joining forces with Mr. Trump? What about serving as his running mate?

The caller was Calley Means, a health care entrepreneur who had advised Mr. Kennedy on chronic disease policy. He suggested that it might be a moment for unity — Mr. Trump had just narrowly escaped the same fate that had befallen Mr. Kennedy’s father and uncle. Mr. Kennedy, who was running an independent campaign for president, said he wasn’t interested in the vice presidency, and the call ended.

A short while later, Mr. Kennedy called back. Yes, he said, he would speak with Mr. Trump.

The calls set off a frenzy of calculations and soul-searching inside the Kennedy camp: What, if anything, was on the table? Could an alliance with Mr. Trump give Mr. Kennedy more power to address issues he had described throughout his campaign — chronic disease, censorship, corporate reach into government agencies, the war in Ukraine? Or would it tear apart his coalition, and his family?

Mr. Trump was not, at that point, seriously considering adding Mr. Kennedy to the ticket. Still, Mr. Means’s efforts presented an opening to bring Mr. Kennedy into the fold and remove him as a potential drain on Mr. Trump’s votes.

What followed was a six-week crush of behind-the-scenes discussions, embarrassing missteps, secret meetings and private misgivings, culminating in Mr. Kennedy’s suspending his campaign and backing Mr. Trump.

The Trump-Kennedy alignment, one of the strangest in modern political history, brought together two men of extraordinary ego and unpredictability. Each candidate, a Republican and a former Democrat, had publicly disparaged the other during the campaign. Now, each had made the political calculation to embrace the other.