Cooper Tire Europe (CTE) said it had begun a consultation about ceasing light vehicle tyre production at its factory in Melksham, Wiltshire, which employs 732 people.
The business is part of the US-based Cooper Tire & Rubber Company, which in 1997 took over the Avon Tyres site.
Avon has been in Melksham since 1890 and boasts of its history supplying brands such as Aston Martin, Rolls-Royce, Bentley and Land Rover in past years.
Its motorsport division, which still supplies race championships around the world, was part of Aston Martin's victory in the world sports car championship in 1959.
CTE said it would continue to make specialised motorsports and motorcycle tyres at Melksham but other tyres will now be made at "other sites within the broader global manufacturing footprint" of the wider Cooper group.
Cooper has factories in Serbia, the US, Mexico and China.
The company said the move could see 300 redundancies over a 10-month period. Melksham would remain CTE's headquarters and sales and marketing office.
It said the site in its present form was "not competitive within the Cooper global manufacturing network or the tyre industry at large".
CTE general manager Jaap van Wessum said: "We know the prospect of making positions redundant is difficult for our colleagues and the local community to consider.
"Yet, for Cooper Tire Europe to thrive into the long-range future, remaining a large Melksham-based employer, and meeting our obligations, we must explore obtaining light vehicle tyres from locations other than Melksham.
"It is essential for Cooper to be globally competitive in the tyre industry
"To deliver on our strategic growth objectives, we must produce quality light vehicle tyres in high volumes and at a competitive cost.
"Unfortunately, the Melksham site is an older, smaller facility that does not offer economies of scale and it is the highest cost facility in the global Cooper network.
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"Due to facility age and location in the centre of town, it is our current view, subject to consultation, that it is not economically feasible to modernise or expand there for light vehicle tyre production."
Tony Hulbert, regional officer at trade union Unite, said: "These proposed job losses will be a hammer blow for the Coopers workforce and the community, sending shockwaves through the local economy."