Nearly 1,800 are employed at the Stafford Road plant
The company warned that the industry was in the grip of problems caused by "severe over-capacity".
The Stafford Road plant in Wolverhampton has been producing tyres since the late 1920s.
If the jobs go they will be lost from Goodyear's truck tyre division.
'Plague or cholera'
Unions say they are urgently seeking talks with management of the US-based company.
In January the labour force at the Stafford Road plant rejected a deal under which 300 jobs would have gone in exchange for workers accepting a pay cut and longer hours.
At the time, Roger Shutt, industrial organiser for the Transport and General Workers' Union (TGWU) said the decision was like asking his members to choose "the plague or cholera".
On Thursday the company's human resources director, Martin Smith, said: "The recent sharp downturn in demand due to the overall economic environment puts huge pressure on us."
About 1,800 workers are employed at the plant producing car, truck and agricultural tyres.
TGWU national officer Peter Booth said: "The potential loss of over 450 jobs and the part closure of the site is not a good starting point but we will do everything in our power to safeguard tyre-making in all parts of the Goodyear operations."
In 1999 Goodyear took a 10% stake in Sumitomo surpassing its rivals Bridgestone and Michelin, to become the world's largest tyre manufacturer.