David Zindel, Paul's son, told The Guardian that the film, a lyrical drama about a mute cleaning lady's unique bond with a sea creature, has been "obviously derived" from his father's play 'Let Me Hear You Whisper'.
del Toro, who directed and co-wrote the film with Vanessa Taylor, is yet to respond on the allegations.
Zindel's play is about a female janitor in a research laboratory who bonds with a captive dolphin and tries to rescue the creature.
Fox Searchlight, which distributed the film, denied the accusations in a statement.
"Guillermo del Toro has never read nor seen Mr Zindel's play in any form. Mr. del Toro has had a 25-year career during which he has made 10 feature films and has always been very open about acknowledging his influences," it said in the statement.
"If the Zindel family has questions about this original work we welcome a conversation with them," it added.