The winger who missed out on World Cup selection and who has been criticised by senior members of the German squad recently met with coach Joachim Löw on Friday and then sparked a Twitter avalanche when he quit the squad.
Löw said Sané had withdrawn from preparations for their game with Peru “for private reasons”.
German daily Bild reported on Saturday Sané’s partner Candice Brook had given birth late on Friday, with reaction from Sané’s father.
“My granddaughter was born yesterday (Friday), we’re very happy, her name is Rio Stella,” Souleyman Sané told Bild.
Sané, 22, missed Germany’s disastrous World Cup campaign in Russia but was brought back by Löw into his first squad since the tournament.
He came off the bench towards the end of Thursday’s 0-0 draw with France in Munich in their first game in the inaugural UEFA Nations League.