Weighing three-and-a-half kg, the baby, born May 21 at Mumbai's Lilavati hospital, is Basu's second child.
"Anurag Basu is delirious after the birth of his second daughter. He hasn't named yet because he is still looking for one," said a source close to the director.
His elder daughter was born after the release of his successful film "Gangster - A Love Story", which he directed after recovering from acute leukemia.