It is the first time the world's biggest long-haul carrier has disclosed how many jobs will be lost.
Prior to the crisis, Emirates had 60,000 staff.
Sir Tim Clark said the airline had already cut a tenth of its staff but said: "We will probably have to let go of a few more, probably up to 15%."
The global airline industry has been severely impacted by coronavirus, with activity all but grinding to a halt.
In an interview with the BBC, Sir Tim said Emirates was "not as badly off as others".
But its current situation marks a steep turnaround in the fortunes of the airline, which he said before the pandemic was "heading for one of our best years ever".
The job cuts sweeping the wider aviation industry are fuelling concern amongst Emirates staff that things might get worse.