The majority of the pieces that make up the new museum's collection were recovered during seizures led by the Manhattan District Attorney's Antiquities Trafficking Unit.
The British Museum and the V&A said they were researching the artifacts in question.
The museum's director said that the decision to give the sculpture back was a "collaborative one."
Mexican politicians have been calling on auction houses around the world to cancel sales of artifacts.
"Human remains from colonial contexts have no place in our museums and universities," said Germany's culture minister.
The precinct will include a national resting place for ancestral remains repatriated from overseas collections.
Half the surviving marbles taken from the Parthenon temple in Athens are held by the British Museum.
Together, they are valued at a combined $10 million.
The objects came into Houston in 2009, though a civil war prevented most from returning home immediately.
Amid pressure from Nigerian officials, the museum has officially given back three objects.