The UK engineering firm, with plants in Bristol and Cowes, Isle of Wight, is ready to pay £800m ($1.3bn) for the business, says the Sunday Times.
Volvo Aero makes aircraft engines and components, including the jet engine used in the Saab Gripen combat jet.
GKN's aerospace business is the firm's second biggest division. It makes parts for civilian and combat aircraft as well as helicopters.
Volvo said in November that it would sell its aerospace unit to focus on trucks and construction equipment. It left car production more than a decade ago when it sold Volvo Cars to Ford.
Germany's MTU Aero Engines was in talks to buy the business but reports say it dropped out last month.
In reaction to the story, GKN says that it "does not comment on market speculation".
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