Jeff Bezos, the founder of Amazon and the owner of The Washington Post, did not become the world’s richest man by refusing to do what it takes to win. And now that he finds himself in a mud fight with the supermarket tabloid that exposed his extramarital affair, he is showing off the same drive that helped him turn a simple idea — selling books on the internet — into an all-purpose, $780 billion powerhouse.

Mr. Bezos has risked significant personal embarrassment in taking on American Media Inc., the company that owns The National Enquirer, which last month devoted 11 pages to the tale of his extramarital affair. But with a personal fortune of more than $130 billion, he has the means to torment his tormentors. And as he showed Thursday with his surprise blog post, which accused the tabloid publisher of “extortion and blackmail,” Mr. Bezos is willing to get dirty in the pursuit of victory.

He has attacked American Media right where it hurts — its shaky legal position — and the strategy is showing early signs of paying off.

Federal prosecutors are reviewing Mr. Bezos’ claim that he has been extorted, according to two people briefed on the matter who were not authorized to discuss it publicly. And those prosecutors have planned a meeting with Mr. Bezos’ representatives, one of those people said.

If American Media is found to have broken a law — any law — it would be in violation of a deal with federal prosecutors from the Southern District of New York. The agreement was struck in September after American Media admitted paying hush money during the 2016 presidential campaign to protect Donald J. Trump from allegations of an affair. Under the deal, the company would not be prosecuted for its Trump-related efforts as long as it stayed out of legal trouble for the next three years.

Even as American Media fell into legal jeopardy last year, its board stood by the company’s chairman, David J. Pecker, and its top news executive, Dylan Howard. On Friday, however, the company announced that the board was starting an investigation of Mr. Bezos’ claims.

“American Media believes fervently that it acted lawfully in the reporting of the story of Mr. Bezos,” the company said in a statement. “Nonetheless, in light of the nature of the allegations published by Mr. Bezos, the board has convened and determined that it should promptly and thoroughly investigate the claims. Upon completion of that investigation, the board will take whatever appropriate action is necessary.”

[Read a primer on the players and plot points in the fight that has erupted between Mr. Bezos and American Media.]

The Enquirer exposé detailed Mr. Bezos’ affair with Lauren Sanchez, a former host of the Fox show “So You Think You Can Dance.” It appeared a day after Mr. Bezos announced on Twitter that he and his wife of 25 years, the novelist MacKenzie Bezos, were getting divorced.

Mr. Bezos wrote the blog post himself, according to one person close to him, who was not authorized to speak publicly, and several lawyers reviewed it before he hit publish. In the post, he said American Media had threatened to publish compromising photographs of him, including a “below the belt selfie,” if he did not publicly affirm that The Enquirer’s reporting on his affair with Ms. Sanchez was not motivated by political concerns.

“If in my position I can’t stand up to this kind of extortion, how many people can?” he wrote.

At 55, Mr. Bezos has become a prominent player in Hollywood and Washington. Thanks to Amazon’s entry into the entertainment business, he is no stranger to the red carpet and Golden Globes after-parties. And with a 27,000-square-foot mansion in the capital’s Kalorama neighborhood, which he bought for $23 million three years after he acquired The Post in 2013, he is a figure to be reckoned in political circles, albeit one who suffers name calling from the president’s Twitter account.

Now, in his fight, Mr. Bezos has assembled an all-star crisis-management team of protectors and lawyers who have popped up alongside powerful political figures and celebrities over the last few decades.

A key team member is the veteran Hollywood lawyer Martin Singer, who is technically representing Mr. Bezos’ longtime security chief, Gavin de Becker. Mr. Singer knows exactly how American Media works, and he has not always worked on the virtuous side of the street.