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The local train services are also affected in Mumbai due to reduced visibility caused by winds and heavy rains.
Flyers can explore a range of options to reschedule their travel plans.
Air India Express said its website, reservation and airport check-in systems were also impacted.
Other carriers like SpiceJet and Akasa Air also saw disruptions in their online check-in services.
During this time booking, check-in, access to your boarding pass, and some flights may be impacted.
Flight departures at T1, which has only domestic flight operations, have been suspended till 2 pm.
If the passengers are planning to stay with their friends or relatives in the UAE, their addresses and phone numbers should be produced.
The bomb threat was received as flight 6E 5314, which departed from Chennai for Mumbai.
Besides Delhi, air taxi services would be launched in Mumbai and Bengaluru in the initial phase.
The directive was issued on February 16 to seven airlines -- Air India, IndiGo, Akasa Air, SpiceJet, Vistara, AIX Connect and Air India Express.