"NEWS NEWS NEWS !! I AM GOING TO BECOME A GRANDFATHER .. AISHWARYA EXPECTING .. SO HAPPY AND THRILLED !!!," Bachchan wrote on his Twitter page late Tuesday.
Aishwarya Rai, 37, married Bachchan's actor son Abhishek, 35, in April 2007. The child will be their first and comes after weeks of speculation in Indian gossip columns about the actress' fluctuating weight.
Bachchan senior later said on the micro-blogging site that he was "overwhelmed" with the response to the news from followers and had received 2,843 tweets of congratulation in just half an hour.
Writing on his blog bigb.bigadda.com, he said: "There is joy and happiness around us."
There was no immediate indication when the baby was due.
The Bachchans -- Amitabh, his actress wife Jaya, Aishwarya and Abhishek -- are Bollywood's "first family" and have a fanatical following in India and around the world. His daughter, Shweta Nanda, has two children.
The family patriarch said he was particularly pleased with the news because "now cometh the progeny of my son that shall bear the surname Bachchan - the first in this generation...".
The family boycotted the annual Filmfare magazine awards in 2010 after a tabloid newspaper in Mumbai wrote that Rai could not have children, prompting a legal notice to be sent demanding a retraction and an apology.
Rai, who took the Miss World crown in 1994, made her debut in Hindi-language Bollywood in the late 1990s and has gone on to star in more than 40 Indian films.
Most recently she was in the blockbuster South Indian film "Endhiran" (Robot) alongside Tamil-language superstar Rajinikanth, which broke box office records in India.
She appeared as a jewel thief in Steve Martin's "The Pink Panther 2". Her other English-language films include "Bend It Like Beckham" director Gurinder Chadha's "Bride and Prejudice" in 2004 and "The Last Legion" in 2007.
She was recently seen at the Cannes film festival promoting her upcoming film "Heroine", which was due to begin shooting this year.
The chief executive of UTV Motion Pictures, Ronnie Screwvala, said at the time that Rai had star power.
"She is the brand ambassador for India and for Indian cinema when it comes to the French Riviera."
Rai also has a number of lucrative modelling contracts, including for French cosmetics firm L'Oreal.
Source : Yahoo