Arozarena allegedly tried to take his daughter from her mother and assaulted the woman's father, sources told Jeff Passan of ESPN.
Major League Baseball is investigating.
Any discipline from the league would fall under the league's domestic violence policy, Passan notes.
Abel Guerra, Arozarena's agent, confirmed his client was arrested after an "apparent domestic dispute" involving his daughter, according to John Romano of the Tampa Bay Times.
"When it comes to family issues there are always layers of complexity. Everything is perspective," said Guerra, who had spoken to members of the family but not Arozarena. "Hopefully everything clears itself up. It's a horrible, unfortunate situation."
The 25-year-old has not commented about the arrest and is expected to be in custody for 48 hours.
The Rays released a statement on Tuesday night.
"We are aware of reports that Randy Arozarena was detained in Mexico," the statement read, according to Juan Toribio of MLB.com. "We are gathering more information on this situation, and we have no further comment at this time."