BBC | Officials from the government's Directorate General of Civil Aviation and Bureau of Civil Aviation Security did not respond to the BBC's emails for comment.
Apart from Air India, IndiGo, SpiceJet and Akasa Air flights also received threats.
On Monday, three international flights that took off from Mumbai were diverted or delayed after an X (formerly Twitter) handle posted threats. Police have detained a teenager in connection with this.
On Tuesday, seven flights, including the two Air India planes, were affected by the threats issued by another X handle which has now been suspended. Screenshots of some of the posts show the user had tagged the airline and local police and mentioned the flight number.
Air India said in a statement that it was co-operating with authorities to identify the people behind the threats and would consider legal action to recover damaged incurred.
Every Indian airport has a Bomb Threat Assessment Committee which assesses the gravity of the threat and takes action accordingly. A threat can lead to the involvement of the bomb disposal squad, sniffer dogs, ambulances, police and doctors.
Passengers are off-loaded from the plane along with cabin baggage, check-in baggage and cargo, and they are all screened again. Engineering and security teams also search the plane before it is cleared for flying again.
The resultant delay can cost thousands of dollars in damages to airlines and security agencies.
For flights heading for other countries, it can also lead to international agencies getting involved, like in Singapore and Canada.
On Tuesday, Singapore's defence minister said that two of the city state's fighter jets "scrambled and escorted" the Air India Express plane before it landed safely at Changi airport. The plane was flying from Madurai in India to Singapore.
"Once on the ground, the plane was handed to the Airport Police. Investigations are ongoing," Ng Eng Hen wrote.
The aircraft later landed safely at Changi.
In Canada - where the Air India flight to Chicago had landed at Iqaluit airport as a precautionary measure - the Royal Canadian Mounted Police said it was investigating the threat.
Air India said on Wednesday that a Canadian Air Force plane was taking the passengers to Chicago. It's not clear yet when the Air India plane will be allowed to take off.