Apple Computer started the first iTunes music store in 2003 and is now the market leader for music downloads but Apple Corps wants to reinstate the 1991 deal and claim financial damages.
Judge Martin Mann will hear the case at the Royal Courts of Justice in London, having turned down an application by Apple Computer to have the case heard in California back in 2004.
Apple Corps, which is owned by former Beatles Paul McCartney, Ringo Starr and the widows of John Lennon and George Harrison, was founded in 1968, whilst Apple Computer, the firm whose home computers helped start the personal computer industry, was created in 1976.
Apple Computer's logo is an apple with a neat bite out of the side and a whole green Granny Smith apple represents the Apple record company.
An agreement between the two with regard to sharing use of the Apple trademark was first established in November 1981, however, as Apple Computer's business increasingly went into the world of entertainment, the company sought a less restrictive trademark agreement, and a court battle began in 1989.
Details of the eventual deal, thrashed out in London's High Court over two years, were never revealed but Apple Corps were thought to have emerged with about $30m (£17m) from Apple Computer.
