The former says it is due to capacity issues at its Kuala Lumpur hub, while the Indian carrier is mulling the move due to its poor financial state.
Malaysia-headquartered low-cost airline group AirAsia plans to delay eight of the 24 Airbus A320s it is due next year.
The delivery slip is being sought because of airport terminal space limitations arising from the delay in getting a new low-cost carrier terminal at Kuala Lumpur international airport, says the airline.
It is unclear if the deal has been agreed with Airbus, or how far back the deliveries have been pushed.
According to Flight's ACAS database, the group has 115 A320s on order. It currently operates 76 aircraft - mostly A320s - and will phase out its remaining Boeing 737-300s next year.
AirAsia has complained that the current low-cost carrier terminal at Kuala Lumpur is too small. The government has announced that a new one will be built, but it is unclear when it will be completed.
Meanwhile, Indian civil aviation minister Praful Patel has warned the country's parliament that the "rescheduling or cancellation of future aircraft deliveries" is one of the measures that Air India plans to adopt to "improve its financial position".
The loss-making flag carrier is seeking to restructure its business in order to get financial aid from its owner, the Indian government. It is seeking an equity infusion of Rp12.31 billion ($254 million) and a soft loan of Rp27.5 billion that will be repaid over 15 years.
It has taken delivery of 50 of the 111 Boeing and Airbus passenger aircraft ordered in 2005. As a result, its debt swelled to Rp152.41 billion in June after paying for new aircraft.
Patel, who also told the parliament that Air India lost about Rp72 billion in the year ended 31 March, said that the carrier was taking other steps to stem losses.
These include cutting flights on traditionally loss-making routes, returning leased capacity, reviewing manpower requirements and cutting the head count.