The company, whose Scottish operation includes work at Edinburgh, Glasgow, and Aberdeen airports, told staff on Wednesday that up to 4,556 jobs could be cut - more than half its UK workforce.
Chief executive Jason Holt said the company had to reduce the size of its workforce to survive.
The collapse in air travel means its revenue is forecast to be almost 50% lower than last year.
It comes after news that 200 Swissport jobs in Scotland were put at risk in March, after the collapse of airline Flybe.
Mr Holt said in a message to staff: "We must do this to secure the lifeline of funding from lenders and investors to protect as many jobs as possible in the United Kingdom and Ireland.
"It's true that we've seen tough times before - volcanic cloud, 9/11, the financial crisis - and we've weathered these. But this time it's different. We have never seen anything like Covid-19 in our lifetimes.
"We are now facing a long period of uncertainty and reduced flight numbers, along with significant changes taking place to the way people travel and the way goods move around the world. There is no escaping the fact that the industry is now smaller than it was, and it will remain so for some time to come."
Swissport employs 8,500 workers at airports, including baggage handlers and check-in staff.
Unions said the announcement was "devastating" news, especially for regional economies, and highlighted their call for the Government to offer help to the aviation industry.
Nadine Houghton, national officer of the GMB, said: "With Swissport now considering job cuts on this scale we have deep concerns about the viability of many of our regional airports and the benefits for regional connectivity that they bring."
Oliver Richardson, national officer of Unite, said: "We can't wait any longer, the UK Government needs to urgently intervene with a bespoke financial package and an extension of the 80% furlough scheme for the aviation industry.
"Speed is of the essence if the Government is to save thousands of aviation jobs and livelihoods."
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