The store at No.1 Piccadilly has been a record store since Tower Records opened there in 1984. It was acquired by Virgin in 2003 and subsequently rebranded as Zavvi four years later.
Primark is believed to be looking at a number of Zavvi sites including the home entertainment retailer's other flagship store at the eastern end of Oxford Street at the corner with Tottenham Court Road.
If Primark secures the two sites it will have a triangle of flagship-sized stores in London's West End. The chain already owns a store on Oxford Street, which it acquired from Debenhams in 2007.
This week Primark announced postive sales growth, including new store openings, of 18% for the 16 weeks preceding January 3.