Franco did not play in Sunday's 9-2 loss to Cleveland.
"During today's game, we were made aware of the social media posts that are circulating regarding Wander Franco," the Rays said in a statement. "We take the situation seriously and are in close contact with Major League Baseball as it conducts its due diligence."
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Tampa Bay did not detail the nature of the social media posts.
Osleivis Basabe started at shortstop on Sunday in the 22-year-old's major league debut. Basabe, who went 1 for 3 with a double, learned he was being brought up during his game Saturday night with Triple-A Durham.
Franco started Sunday's game in the Rays' dugout, then left it in the fifth inning.
Sunday's promotion at Tropicana Field was the Wander Franco Snapback Hat for fans 14 and under.
After the game, Rays manager Kevin Cash was asked whether there was any issue with Franco not playing other than a regular day off. Cash replied: "No."
"I'm aware of the speculation, and I'm not going to comment any further on that," Cash said. "The day off was because (it was) a day off."
Franco was benched for two games in late June by the Rays for how he has responded to frustrating situations and not being the best teammate at times.
Tampa Bay signed the 22-year old to an $182 million, 11-year contract in November 2021, a deal with a team option for 2033 that could be worth $223 million.
Franco, an AL All-Star this season, is hitting .281 with 17 homers, 58 RBIs and is 30 for 40 in stolen base attempts in 112 games.