Confirmation from the Middle Eastern airline is expected soon.
The daily service will end in October, just three years after being launched in competition with Qatar Airways to Doha.
Most passengers have used the hubs as stepping stones to other destinations on the airlines' long-haul networks, such as in Asia and Australia.
However, the airline has a smaller network of destinations than Qatar or Emirates, which flies twice a day between Glasgow and Dubai.
The decision will come as a major blow to Etihad, whose then chief executive James Hogan announced in Scotland in 2014 that it would launch two Scottish routes.
The second one was due to start by 2020.
Passengers booked on flights after October are likely to be offered to be re-routed via another airport, switched to another airline or given a refund.
An aviation source said: "The route closure decision fits with a period of cutbacks Etihad has been making across their network and a relatively poor performance at Edinburgh..
"Qatar Airways will be upgrading to larger Airbus A350 from Edinburgh at the same time.
"The new China flight between Edinburgh and Beijing from June won't help Etihad’s economics in the route either."
Aviation analyst John Strickland, of JLS Consulting, said: "Etihad is in a major phase of retrenchment, having exited several of its investments including Air Berlin and Alitalia, and also divesting of aircraft and closing routes.
"Emirates also continues to perform strongly on its Glasgow to Dubai service.
"The Edinburgh global market is a large one but the pace of competition has increased to the Gulf and Asia.
"Now, the Abu Dhabi route is paying the price of this and Etihad's wider strategic review as it stems its losses."
Ken McLeod, president of the Scottish Passenger Agents Association, which represents travel agents, said: “We are very disappointed that any airline has to pull a route, but no-one underestimates the challenge of Etihad, Emirates and Qatar sustaining four flights a day from Scotland to worldwide destinations via the Middle East.
“The introduction of a direct China flight with Hainan this summer won’t have been the main reason why Etihad may pull out, but it won’t have helped.
“If Scotland loses Etihad we will be very sorry, because there has been a lot of hard work put in over the last three years.
“However, Edinburgh has a huge number of destinations in its portfolio, and will continue to grow and prosper with its wide variety of carriers.”
Etihad did not deny the route would be closed.
Its spokesman said: “Etihad Airways continues to review its network and make a select number of adjustments which will be announced at the appropriate time, taking into account the needs of our staff, partners, and customers.
"In recent weeks the airline has increased its European schedule, introducing an additional daily flight from Rome to Abu Dhabi and launching services from Baku.”