The Uruguayan midfielder reacted angrily to alleged derogatory remarks that Baena had made about his unborn child, sources said.
Valverde waited by the Villarreal team bus in the stadium's parking lot after the match to confront Baena and asked him to repeat what he claimed had been said on the pitch.
Valverde's partner Mina Bonino revealed on social media in February that the couple had believed they had lost their child, before further tests revealed that the pregnancy was in fact healthy.
Sources told ESPN that Real Madrid and Valverde say Baena first made the comments when the two teams met in the Copa del Rey on Jan. 19 -- a game that both players started, with Madrid coming back from 2-0 down to win 3-2 -- saying "cry, because your child won't be born."
Villarreal and Baena have not confirmed that claim, but sources said they believed Valverde's actions on Saturday were unacceptable.
"Fede Valverde, loutish and cowardly," the Villarreal team delegate, Xisco Nadal, tweeted on Saturday.
Villarreal have spoken to Spanish police about possible courses of action, sources said, and Baena will decide if he wants to press charges.
Valverde, 24, joined Madrid in 2016 and has become a key player in recent seasons.
The club are set to face Chelsea in their Champions League quarterfinal first leg on Wednesday.
Baena, 21, is a Villarreal youth product who has become a first-team regular this season, scoring four goals in 25 LaLiga appearances.