The issue which occurred in the early hours of Sunday (23 June) has particularly affected services in Terminals 1 and 2, the airport said.
Passengers due to fly from these terminals have been advised to not to come to the airport until further notice and to contact their airlines.
Update: MAN flights resume after major power failure
Images and footage posted on social media show passengers standing by baggage carousels in near darkness.
An airport spokesperson said: “We are working to get passengers at the airport onto their flights. The disruption caused by the outage means in some instances baggage may not be on those flights.
"We will work with airlines to make sure passengers are reunited with any baggage that is not on their flight as soon as possible.
"A number of arriving flights are being diverted to other airports so anyone due to pick up passengers from the airport are advised to check before they travel.
"We apologise for any inconvenience and aim to restore normal service as soon as possible.”
Aer Lingus, Air Transat, easyJet, Emirates, Etihad Airways and Finnair are among the airlines that operate from Terminal 1.
The airlines which fly from Terminal 2 include Aegean Airlines, Air France, Cathay Pacific, Ethiopian Airlines, Jet2, KLM and Norwegian.
A Jet2 spokesperson said: "We would like to make you aware that there is a power outage that is affecting all terminals at Manchester airport.
"For flights departing from Manchester today (23 June), we are expecting delays during check-in and security screening at the airport.
"All Jet2.com flights are due to operate as scheduled. Our dedicated operations teams are working hard behind the scenes at our UK-based HQ to get you on your way as soon as possible. We apologise for any inconvenience this may cause."
The airport said passengers flying from Terminal 3 should come to the airport as normal unless advised otherwise by their airline but “could be affected by delays”.
