Ginobili, on his Twitter page, announced Sunday's births of Dante and Nicola.
Ginobili says mother and boys are doing great, and he later added "I just changed my 1st diaper!"
The Spurs were swept in the second round of the playoffs by the Phoenix Suns.
Ginobili in April announced he would not play for his native Argentina this summer in the world championships in Turkey because of his pending parenthood, plus he wants to stay healthy.
Ginobili spent the past two summers recovering from ankle injuries.