A few months ago, a three-post exchange between Jeff Bezos and Elon Musk on Mr. Musk’s X would have passed for petty sniping between billionaire rivals.

But times have changed.

“Just learned tonight at Mar-a-Lago that Jeff Bezos was telling everyone that @realDonaldTrump would lose for sure, so they should sell all their Tesla and SpaceX stock,” Mr. Musk wrote Wednesday night, referring to two of his companies. He added an emoji for a snickering face, with a hand covering the mouth.

“Nope. 100% not true,” Mr. Bezos responded on Thursday morning.

“Well, then, I stand corrected,” Mr. Musk wrote back, with a laughing-crying emoji.

With President-elect Donald J. Trump's history of animosity toward Mr. Bezos, the posts carried an unspoken message about Mr. Musk’s growing power within the incoming administration.

The exchange — brief, brassy and fairly typical of Mr. Musk’s overwhelming presence on X — could foreshadow a bumpy next few years for Mr. Bezos and the companies he started, Amazon and the rocket maker Blue Origin. It was also a reminder that the power dynamics in the longtime rivalry between the world’s two richest men changed on Nov. 5.

Plenty of tech executives have drawn Mr. Trump’s wrath over the last few years. Perhaps none more than Mr. Bezos, largely because he owns The Washington Post, which has frequently written critically about Mr. Trump. (The Post did not endorse a presidential candidate this year, a decision that angered many of its readers and that Mr. Bezos publicly defended.)