"If I have a boy and a girl I will call them Ash and Abhishek," Gurinder Chadha, referring to Aishwarya's nickname, was quoted as saying in the Times of India.
Chadha is pregnant with twins and is expected to give birth in July.
Rai, a former Miss World, and Bachchan are two of Bollywood's biggest stars and have been dubbed "India's Brangelina" after the famous Hollywood duo.
They got engaged in January, setting to rest feverish speculation in the media about their hush-hush romance.
Chadha, who directed Rai in "Bride and Prejudice", said she had kept her next big project, "Dallas", on hold until the babies are born.
"No film experience can compare with this," she said.
"I wanted to pull out of the project. But the producers insisted that I do it."
The expectant mother was made an officer of the Order of the British Empire (OBE) by Queen Elizabeth this week.