Maradona reacted angrily to comments the 80-year-old Grondona made to local media this week that have been interpreted as a veiled reference to Maradona's much-publicized battle with drugs.
"I'm old, but healthy," Grondona was quoted as saying, noting that others had "created problems" for themselves.
Maradona, who signed a deal to coach Dubai-based club Al Wasl last week, told Sunday's Clarin newspaper he has been clean for seven years and will sue Grondona.
"This time I'm going to step on the accelerator and step on the head of whoever it might be. Everyone, even his grandchildren, will have to submit to the analyses my lawyers are going to ask for," Maradona told Clarin.
"I don't have anything to hide. I''m going to be the first to have the corresponding exam done," Maradona continued.
"Because this is going straight to the authorities. Let everybody know and record it: I'm going after Grondona's family."
The ongoing feud between the two erupted when Grondona decided not to renew Maradona's contract as Argentina manager after the 2010 World Cup.
"It's getting harder and harder for Grondona because of his age," Maradona was quoted as saying. "Hopefully, he will realize and retire."