Amazon’s new NFL deal could be a chance to bring television advertising to the digital age. That is, if the company even wants to.

This week, Amazon announced a one-year, $50 million deal to stream 10 Thursday night NFL games this coming season, taking over from Twitter, which previously held the streaming rights. The arrangement means Amazon is also entitled to sell two minutes per hour of advertising during the games—ad space normally sold by local TV stations carrying the broadcast.

Amazon indicated that it will use some of that time to promote its original series, which have gained steam in recent years with hit shows like “Sneaky Pete” and “Man in the High Castle.”

But there may be a bigger opportunity for the online retailer, which hasn’t said much else about the advertising angle of the deal.

Amazon has a history of quietly nudging into sectors of the advertising business before ramping up its ambitions. For example, after building out ad technology primarily for its own use, the e-commerce giant has in recent years become a significant player in the programmatic display advertising business. Recently, Amazon also has entered the lucrative search business while building out a suite of ad technology products designed to help web publishers make more money.

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Amazon is often referred to as a “sleeping giant” in the ad world. Barclays estimates the company’s advertising revenue could expand quickly to cross the $2 billion mark next year.

“I think they have an enormous advertising business in front of them,” said Michael Bologna, president of One2one Media, an company that focuses on advanced forms of TV advertising. Amazon, run by Chief Executive Jeff Bezos, was probably purposefully quiet about the ad opportunities with the NFL deal, he said. “The question is when he unleashes it.”

The live streams of the NFL games will be available to Amazon Prime members, though the matchups will still be shown on regular television by networks like CBS and NBC. For Amazon, driving Prime adoption and retention is likely the focus of the deal more so than generating ad revenue, Barclays analysts said in a note this week. Analysts have estimated that Amazon has more than 60 million Prime subscribers world-wide.

But if executives are so inclined, Amazon could use the football games to experiment with presenting different ads to different consumers at the same time, which could theoretically represent the future of web-delivered TV advertising.

“It could be a whole new level of addressability,” said Mr. Bologna. “They’d know I am an NFL fan, that I drink Pepsi and eat Doritos.”

Because the games will only be available to Amazon Prime subscribers—who pay $99 a year for free, two-day shipping and access to music, movies and TV shows—viewers will likely need to be logged into their accounts to watch.

There could be opportunities beyond just more dynamic TV ads, said Seth Walters, a senior partner at Modi Media, an ad buying division focused on new forms of TV advertising within GroupM. He theorized that Amazon could enable advertisers to run ads during NFL games designed to prompt viewers to take actions using Amazon’s voice-controlled home assistant Alexa, such as ordering a pizza.

Plus, after the fact, Amazon could theoretically show specific ads to Amazon.com AMZN -2.43% shoppers based on the fact they have watched NFL games, Mr. Walters said.

But it’s what Amazon could learn about TV advertising that has the biggest potential ramifications for the marketing world.

“Even though it’s only a single season experiment, it lays the foundation for advanced advertising and serves as a battle cry, claiming a leadership position for the future of pay TV,” said Mr. Walters. “If this leads to a multiyear renewal with the NFL, more deals for live programming or the introduction of an Amazon skinny bundle, it’s game over.”

Or game on.

Write to Mike Shields at mike.shields@wsj.com