A grand jury has not yet heard the manslaughter case against the dad whose twin babies died after he left them in a hot car all day, prosecutors said Thursday.

Juan Rodriguez appeared in Bronx criminal court for a brief hearing in which prosecutors told a judge that there has been “no grand jury action” taken at this time, Bronx District Attorney’s Office spokeswoman Patrice O’Shaughnessy told The Post.

“We’re still investigating,” she said.

Rodriguez, who is out on $100,000 bail, remains charged with two counts of manslaughter and two counts of criminally negligent homicide in the July 26 deaths of 1-year-olds Luna and Phoenix. If the case is presented, a grand jury ultimately could decide to indict the Iraq war veteran. Prosecutors have six months to present a case.

Outside the courthouse, Rodriguez’s attorney was optimistic that DA Darcel Clark will eventually dismiss the charges.

“We understand that they will continue to evaluate all aspects of the case and we welcome their evaluation of that,” said the lawyer, Joey Jackson. “We believe at the conclusion of them evaluating all the facts, all evidence and everything in this case, they will come to the conclusion that this is a horrible tragedy.”

Next to Jackson was a weeping Rodriguez, holding his 4-year-old son Tristan and flanked by his wife.

Jackson said the family is “completely crushed by this incident.”

“He has nothing at all to harbor and hide, other than to feel misery and sorrow about what happened in this case,” the lawyer said.

The babies were found unconscious and foaming at the mouth in their car seats inside Rodriguez’s sweltering Honda Accord in the Bronx — after he forgot about them and went to work for eight hours.

His next court date is Aug. 27.

“The purpose of that proceeding will again be to determine whether or not a grand jury has been convened to indict Mr. Rodriguez. We are hopeful that will not be the case,” said Jackson.

Rodriguez’s stepfather, Carlos Hernandez, said the parents don’t have the heart to tell Tristan the truth about what happened to his siblings.

“Tristan is missing his baby brother and sister. He has been asking about them. He tells his mommy he wants them,” said Hernandez. “They have to find a way to explain it to him. He knows something happened. He knows they are not coming home but he doesn’t know why.”

Additional reporting by Kevin Sheehan and Georgett Roberts